May 2007Computer Arithmetic, Number RepresentationSlide 1 Part V Real Arithmetic.
Some arithmetic groups that cannot act on the...
Transcript of Some arithmetic groups that cannot act on the...
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Some arithmetic groups thatcannot act on the line
Dave Witte Morris
University of Lethbridge, Alberta, Canadahttp://people.uleth.ca/�dave.morris
joint with
Lucy Lifschitz, University of Oklahoma
Vladimir Chernousov, University of Alberta
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 1 / 16
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Transformation groups
Given: group — ,
(connected)
manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 3: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/3.jpg)
Transformation groupsGiven:
group — ,
(connected)
manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 4: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/4.jpg)
Transformation groupsGiven: group — ,
(connected)
manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 5: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/5.jpg)
Transformation groupsGiven: group — ,
(connected)
manifold M .
¿ What are the actions of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 6: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/6.jpg)
Transformation groupsGiven: group — , (connected) manifold M .
¿ What are the actions of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 7: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/7.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿
What are the actions of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 8: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/8.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions
of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 9: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/9.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of —
on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 10: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/10.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 11: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/11.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.:
¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 12: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/12.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos
� : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 13: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/13.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � :
— ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 14: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/14.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : —
! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — !
Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 16: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/16.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 17: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/17.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question
¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 18: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/18.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿
9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 19: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/19.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9
action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 20: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/20.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 21: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/21.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (nontrivial) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 22: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/22.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 23: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/23.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest case
dimM � 1, so M � S1 or R.Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 24: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/24.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM �
1, so M � S1 or R.Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 25: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/25.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1,
so M � S1 or R.Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 26: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/26.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so
M � S1 or R.Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 27: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/27.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M �
S1 or R.Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 28: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/28.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1
or R.Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 29: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/29.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or
R.Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 30: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/30.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 31: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/31.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume
M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M �
R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
Example
Z acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 35: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/35.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ
acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 36: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/36.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts
on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 37: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/37.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on
R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 38: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/38.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R.
(Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 39: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/39.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x�
� x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� �
x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 41: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/41.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x
�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�
n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n
=) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =)
Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n
� Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n �
Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
![Page 47: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/47.jpg)
Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm
�Tn)
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �
Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?
I.e.: ¿ What are homos � : — ! Homeo��M� ?
Question¿ 9 (faithful) action ?
Simplest casedimM � 1, so M � S1 or R.
Assume M � R.
ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 2 / 16
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Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
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Question¿ 9 (faithful) action of — on R ?
Today:
— is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 53: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/53.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is
an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 54: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/54.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 55: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/55.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
Example
SL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 56: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/56.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z�
does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 57: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/57.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not
act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 58: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/58.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 59: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/59.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof.
"�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 60: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/60.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#
2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 61: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/61.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2
� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 62: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/62.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I.
So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 63: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/63.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So
SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 64: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/64.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has
elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 65: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/65.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 66: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/66.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But
Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 67: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/67.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has
no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 68: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/68.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:
’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 69: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/69.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’
�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 70: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/70.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0�
> 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 71: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/71.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0
=) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 72: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/72.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =)
’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 73: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/73.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0�
> ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 74: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/74.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� >
’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 75: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/75.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0�
> 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 76: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/76.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0
=) ’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 77: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/77.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =)
’3�0� > 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 78: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/78.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0�
> 0=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 79: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/79.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 80: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/80.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=)
. . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 81: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/81.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . .
=) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 82: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/82.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =)
’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 83: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/83.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0�
> 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 84: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/84.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 85: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/85.jpg)
Question¿ 9 (faithful) action of — on R ?
Today: — is an arithmetic group
ExampleSL�2;Z� does not act on R.
Proof."�1 00 �1
#2� I. So SL�2;Z� has elt’s of finite order.
But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0
=) . . . =) ’n�0� > 0.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 3 / 16
![Page 86: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/86.jpg)
Example:
SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 87: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/87.jpg)
Example: SL�2;Z�
does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 88: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/88.jpg)
Example: SL�2;Z� does not act on R
because it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 89: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/89.jpg)
Example: SL�2;Z� does not act on Rbecause
it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 90: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/90.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 91: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/91.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example
— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 92: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/92.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z�
finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 93: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/93.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp
can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 94: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/94.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be
a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 95: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/95.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.
Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 96: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/96.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has
many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 97: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/97.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 98: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/98.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
Fact
There exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 99: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/99.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist
other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 100: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/100.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples
that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 101: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/101.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.
But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 102: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/102.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are
“small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 103: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/103.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”.
(I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 104: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/104.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think
all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 105: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/105.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known
are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 106: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/106.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in
SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 107: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/107.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 108: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/108.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
Conjecture
Large arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 109: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/109.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups
(R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 110: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/110.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank
> 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 111: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/111.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1)
cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 112: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/112.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 113: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/113.jpg)
Example: SL�2;Z� does not act on Rbecause it has elements of finite order.
Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.
FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).
ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 4 / 16
![Page 114: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/114.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 115: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/115.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume
— is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 116: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/116.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a
“large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 117: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/117.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large”
arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 118: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/118.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:
— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 119: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/119.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É
SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 120: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/120.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z�
= f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 121: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/121.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g
(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 122: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/122.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 123: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/123.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or
— É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 124: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/124.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É
SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 125: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/125.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z
�p
3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 126: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/126.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 127: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/127.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or
. . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 128: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/128.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .
Or — É SL�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 129: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/129.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or
— É SL�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 130: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/130.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z
����
� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 131: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/131.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�
� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 132: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/132.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� =
real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 133: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/133.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real,
irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 134: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/134.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat
alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 135: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/135.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 136: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/136.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But
— � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 137: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/137.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — �
SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 138: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/138.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�,
other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 139: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/139.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other
“small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 140: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/140.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small"
grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 141: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/141.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps.
(Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 142: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/142.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need
rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 143: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/143.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR —
> 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 144: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/144.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 145: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/145.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture
— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 146: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/146.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not
act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 147: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/147.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 148: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/148.jpg)
Question¿ 9 (nontrivial) action of — on R ?
Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)
Or — É SL�2;Z�
p3��
or . . .Or — É SL
�2;Z���
�� = real, irrat alg’ic integer.
But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)
Conjecture— does not act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 5 / 16
![Page 149: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/149.jpg)
Conjecture— no action on R if — É SL�3;Z� or SL�2;Z���
�or . . .
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�or . . .
Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 150: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/150.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�or . . .
Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 151: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/151.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�
Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 152: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/152.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof
combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 153: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/153.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines
bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 154: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/154.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation
and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 155: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/155.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and
bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 156: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/156.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 157: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/157.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups:
U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 158: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/158.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �
"1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 159: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/159.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#,
V �"
1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 160: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/160.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 161: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/161.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 162: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/162.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)
U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 163: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/163.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 164: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/164.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate —
(up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 165: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/165.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V
boundedly
generate — (up to finite index).
— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 166: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/166.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).
— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 167: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/167.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R
=) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 168: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/168.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =)
U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 169: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/169.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 170: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/170.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits
(and V -orbits)
are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 171: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/171.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits (and V -orbits) are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 172: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/172.jpg)
Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z���
�Proof combines bdd generation and bdd orbits.
Unipotent subgroups: U �"
1 �0 1
#, V �
"1 0� 1
#.
Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits (and V -orbits) are bdd.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 6 / 16
![Page 173: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/173.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 174: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/174.jpg)
Bounded generation by unip subgrps
Note:
Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 175: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/175.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix
� Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 176: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/176.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix �
Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 177: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/177.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id
by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 178: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/178.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by
row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 179: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/179.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 180: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/180.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact:
g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 181: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/181.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z�
� Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 182: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/182.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� �
Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 183: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/183.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id
by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 184: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/184.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by
integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 185: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/185.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer
(Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 186: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/186.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z)
row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 187: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/187.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 188: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/188.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example
"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 189: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/189.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#
�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 190: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/190.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 191: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/191.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#
�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 192: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/192.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 193: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/193.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 194: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/194.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 195: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/195.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#
�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 196: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/196.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 197: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/197.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#
�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 198: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/198.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 199: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/199.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 200: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/200.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But
# steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 201: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/201.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps
is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 202: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/202.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is
not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 203: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/203.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:
U and V do not boundedly generate SL�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 204: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/204.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V
do not boundedly generate SL�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 205: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/205.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not
boundedly generate SL�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 206: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/206.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate
SL�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 207: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/207.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 208: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/208.jpg)
Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.
Example"13 315 12
#�
"3 75 12
#�
"3 72 5
#
�
"1 22 5
#�
"1 20 1
#�
"1 00 1
#.
But # steps is not bounded:U and V do not boundedly generate SL
�2;Z�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 7 / 16
![Page 209: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/209.jpg)
Key fact:
g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 210: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/210.jpg)
Key fact: g 2 SL�2;Z� � Id
by integer (Z) row ops,but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 211: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/211.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 212: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/212.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 213: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/213.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark:
In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 214: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/214.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�,
# steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 215: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/215.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded
[Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 216: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/216.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 217: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/217.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)
For Z��� row operations, # steps is bounded.9n, 8g 2 SL
�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 218: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/218.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For
Z��� row operations, # steps is bounded.9n, 8g 2 SL
�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 219: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/219.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z���
row operations, # steps is bounded.9n, 8g 2 SL
�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 220: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/220.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations,
# steps is bounded.9n, 8g 2 SL
�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 221: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/221.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 222: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/222.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n,
8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 223: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/223.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2
SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 224: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/224.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z
����, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 225: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/225.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�,
g � u1v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 226: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/226.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g
� u1v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 227: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/227.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g �
u1v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 228: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/228.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1
v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 229: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/229.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1
u2v2� � �unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 230: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/230.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2
v2� � �unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 231: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/231.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2
� � �unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 232: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/232.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �
unvn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 233: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/233.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �un
vn.I.e., U and V boundedly gen — � SL
�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 234: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/234.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 235: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/235.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e.,
U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 236: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/236.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V
boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 237: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/237.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen
— � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 238: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/238.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 239: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/239.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So
SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 240: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/240.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�
= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 241: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/241.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�=
UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 242: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/242.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 243: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/243.jpg)
Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,
but # steps is not bounded.
Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].
Theorem (Liehl, Carter-Keller-Paige)For Z��� row operations, # steps is bounded.
9n, 8g 2 SL�2;Z���
�, g � u1v1u2v2� � �unvn.
I.e., U and V boundedly gen — � SL�2;Z���
�.
So SL�2;Z���
�= UVUV � � �UV .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 8 / 16
![Page 244: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/244.jpg)
Theorem (Liehl)
SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 245: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/245.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�
bddly gen’d by elem mats.I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 246: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/246.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d
by elem mats.I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 247: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/247.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 248: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/248.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e.,
T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 249: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/249.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id
by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 250: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/250.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by
Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 251: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/251.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops,
# steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 252: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/252.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps
is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 253: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/253.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 254: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/254.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proof
Assume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 255: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/255.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume
Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 256: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/256.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:
8r � �1, perfect square,9 1 primes q, s.t. r is primitive root modulo q:
f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 257: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/257.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r
� �1, perfect square,9 1 primes q, s.t. r is primitive root modulo q:
f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 258: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/258.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1,
perfect square,9 1 primes q, s.t. r is primitive root modulo q:
f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 259: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/259.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 260: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/260.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9
1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 261: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/261.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q,
s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 262: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/262.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t.
r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 263: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/263.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root
modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 264: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/264.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:
f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 265: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/265.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f
r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 266: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/266.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ;
r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 267: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/267.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2;
r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 268: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/268.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3;
: : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 269: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/269.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g
mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 270: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/270.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 271: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/271.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q
�
f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 272: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/272.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 273: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/273.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume
9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 274: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/274.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q
in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 275: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/275.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in
every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 276: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/276.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression
fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 277: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/277.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 278: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/278.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q
� a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 279: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/279.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 280: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/280.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb,
p is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 281: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/281.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb,
p
is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 282: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/282.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb, p is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 283: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/283.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Easy proofAssume Artin’s Conjecture:8r � �1, perfect square,
9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g
Assume 9q in every arith progression fa� kbg.
9 q � a� kb, p is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 9 / 16
![Page 284: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/284.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 285: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/285.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof.
"a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 286: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/286.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#
q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 287: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/287.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime,
p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 288: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/288.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 289: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/289.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 290: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/290.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#
p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 291: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/291.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ �
b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 292: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/292.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�;
p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 293: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/293.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b
� k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 294: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/294.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 295: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/295.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 296: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/296.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#
p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 297: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/297.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit:
can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 298: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/298.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add
anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 299: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/299.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything
to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 300: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/300.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 301: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/301.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 302: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/302.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#
�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 303: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/303.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 304: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/304.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#
�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 305: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/305.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 306: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/306.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 307: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/307.jpg)
Theorem (Liehl)SL�2;Z�1=p�
�bddly gen’d by elem mats.
I.e., T � Id by Z�1=p� col ops, # steps is bdd.
Proof."a bc d
#q � a� kb prime, p is prim root
�
"q b� �
#p‘ � b �mod q�; p‘ � b � k0q
�
"q p‘� �
#p‘ unit: can add anything to q
�
"1 p‘� �
#�
"1 0� 1
#�
"1 00 1
#.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 10 / 16
![Page 308: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/308.jpg)
Bdd generation:
— � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 309: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/309.jpg)
Bdd generation: — �
UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 310: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/310.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 311: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/311.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits:
U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 312: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/312.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits
and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 313: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/313.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits
are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 314: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/314.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 315: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/315.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary
� : — ! Homeo��R�) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
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Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 317: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/317.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
)
every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 318: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/318.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit
on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 319: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/319.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R
is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 320: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/320.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded
) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 321: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/321.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded)
— has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 322: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/322.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has
a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 323: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/323.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 324: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/324.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary
— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 325: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/325.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 326: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/326.jpg)
Bdd generation: — � UVUV � � �UV .
Bdd orbits: U -orbits and V -orbits are bounded.
Corollary� : — ! Homeo��R�
) every — -orbit on R is bounded) — has a fixed point.
Corollary— cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 11 / 16
![Page 327: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/327.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 328: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/328.jpg)
Corollary— cannot act on R.
Proof.
Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 329: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/329.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 330: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/330.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 331: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/331.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has
fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 332: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/332.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 333: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/333.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 334: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/334.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 335: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/335.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 336: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/336.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take
a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 337: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/337.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 338: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/338.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 339: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/339.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on
open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 340: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/340.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval
(� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 341: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/341.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (�
R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 342: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/342.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R)
with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 343: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/343.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with
no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 344: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/344.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 345: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/345.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 346: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/346.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 347: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/347.jpg)
Corollary— cannot act on R.
Proof.Suppose there is a nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
— acts on open interval (� R) with no fixed point.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 12 / 16
![Page 348: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/348.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
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Bounded orbits
Theorem (Lifschitz-Morris)
— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 350: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/350.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— �
SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 351: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/351.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�
1=p��
acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 352: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/352.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�
acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 353: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/353.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R
) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 354: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/354.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R )
every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 355: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/355.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit
bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 356: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/356.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 357: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/357.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �
"1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 358: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/358.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#,
v �"
1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 359: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/359.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 360: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/360.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#,
p �"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 361: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/361.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 362: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/362.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#
Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 363: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/363.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume
U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 364: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/364.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 365: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/365.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x
not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 366: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/366.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd
above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 367: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/367.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit
and V -orbit
of x not bdd above.
Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 368: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/368.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.
Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 369: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/369.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume
p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 370: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/370.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p
fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 371: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/371.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x.
(p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 372: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/372.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts,
so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 373: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/373.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 374: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/374.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog
u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 375: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/375.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.
Then pn�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 376: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/376.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then
pn�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 377: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/377.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 378: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/378.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS
= pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 379: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/379.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS =
pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 380: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/380.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�
� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 381: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/381.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
��
�pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 382: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/382.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�
!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 383: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/383.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!
1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 384: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/384.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�
!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 385: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/385.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!
1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 386: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/386.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 387: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/387.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS
= pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 388: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/388.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS =
pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 389: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/389.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�
� �pnvp�n��x�! 0�x� <1.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 390: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/390.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
��
�pnvp�n��x�! 0�x� <1.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 391: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/391.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�
! 0�x� <1.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 392: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/392.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�!
0�x� <1.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 393: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/393.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x�
<1.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 394: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/394.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <
1.!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 395: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/395.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
!
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 396: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/396.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
! Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 397: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/397.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
! Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 398: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/398.jpg)
Bounded orbits
Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�
�acts on R ) every U -orbit bdd.
u �"
1 u0 1
#, v �
"1 0v 1
#, p �
"p 00 1=p
#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)
Wolog u�x� < v�x�.Then pn
�u�x�
�< pn
�v�x�
�.
LHS = pn�u�x�
�� �pnup�n��x�!1�x�!1.
RHS = pn�v�x�
�� �pnvp�n��x�! 0�x� <1.
! Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16
![Page 399: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/399.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 400: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/400.jpg)
Other arithmetic groups of higher rank
Proposition
Suppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 401: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/401.jpg)
Other arithmetic groups of higher rank
PropositionSuppose
—1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 402: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/402.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 403: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/403.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If
—2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 404: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/404.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2
acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 405: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/405.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R,
then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 406: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/406.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then
—1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 407: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/407.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1
acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 408: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/408.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 409: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/409.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If
—1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 410: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/410.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1
does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 411: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/411.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not
act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 412: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/412.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R,
then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 413: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/413.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then
—2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 414: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/414.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2
does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 415: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/415.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 416: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/416.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods
require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 417: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/417.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require —
to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 418: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/418.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have
a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 419: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/419.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.
Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 420: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/420.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups
are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 421: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/421.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called
noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 422: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/422.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 423: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/423.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)
Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 424: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/424.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse
— is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 425: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/425.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is
a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 426: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/426.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact
arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 427: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/427.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group
of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 428: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/428.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.
Then — :� SL
�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 429: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/429.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then
— :� SL
�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 430: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/430.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
�
SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 431: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/431.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�
or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 432: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/432.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or
noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 433: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/433.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp
in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 434: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/434.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in
SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 435: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/435.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R�
or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 436: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/436.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or
SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 437: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/437.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 438: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/438.jpg)
Other arithmetic groups of higher rank
PropositionSuppose —1 � —2.
If —2 acts on R, then —1 acts on R.
If —1 does not act on R, then —2 does not act on R.
Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :
� SL�2;Z���
�or noncocpct arith grp in SL�3;R� or SL�3;C�.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 14 / 16
![Page 439: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/439.jpg)
Open Problem
Show noncocpct arith grps in SL�3;R� and SL�3;C�cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 440: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/440.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 441: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/441.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 442: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/442.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)
These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 443: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/443.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps
are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 444: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/444.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are
boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 445: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/445.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated
by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 446: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/446.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 447: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/447.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture
implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 448: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/448.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies
no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 449: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/449.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on R
if — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 450: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/450.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif
— noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 451: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/451.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact
of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 452: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/452.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 453: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/453.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case
will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 454: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/454.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require
new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 455: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/455.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 456: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/456.jpg)
Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�
cannot act on R.
Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.
Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.
Cocompact case will require new ideas.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 15 / 16
![Page 457: Some arithmetic groups that cannot act on the linepeople.virginia.edu/~mve2x/Workshop/Slides/morris.pdf · Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on](https://reader030.fdocuments.in/reader030/viewer/2022041016/5ec771aae9b0871e1d6e9f64/html5/thumbnails/457.jpg)
V. Chernousov, L. Lifschitz, and D. W. Morris:Almost-minimal nonuniform lattices of higher rank,Michigan Mathematical Journal 56, no. 2, (2008), 453Ð478.http://arxiv.org/abs/0705.4330
L. Lifschitz and D. W. Morris:Bounded generation and lattices that cannot act on the line,Pure and Applied Mathematics Quarterly 4 (2008), no. 1, part 2, 99–126.http://arxiv.org/abs/math/0604612
D. W. Morris:Bounded generation of SL�n;A� (after D. Carter, G. Keller and E. Paige),New York Journal of Mathematics 13 (2007) 383–421.http://nyjm.albany.edu/j/2007/13-17.html
A. Ondrus:Minimal anisotropic groups of higher real rank,(preprint, 2009, University of Alberta).
D. Witte:Arithmetic groups of higher Q-rank cannot act on 1-manifolds,Proc. Amer. Math. Soc. 122 (1994) 333–340.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 16 / 16