Mapping Class Groups of Infinite-Type...

Mapping Class Groups of Infinite-Type Surfaces Santana Afton and Samuel Freedman The College of William & Mary, University of Michigan

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Transcript of Mapping Class Groups of Infinite-Type...

Mapping Class Groups of Infinite-Type Surfaces

Santana Afton and Samuel Freedman

The College of William & Mary, University of Michigan

Outline

▪ Finite-Type Surfaces

▪ Infinite-Type Surfaces

▪ Symmetries

▪ Symmetries of Surfaces

Finite-Type Surfaces

Sphere Torus 2-holed Torus

Infinite-Type Surfaces

The Flute Surface

Infinite-Type Surfaces

The Flute Surface The Blooming Cantor Tree

What is Symmetry?

Page 10: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

What is Symmetry?

Page 11: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Mapping Classes

Rotation

Page 12: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Mapping Classes

Dehn Twist

Page 13: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Mapping Classes

Handle Shift

Mapping Class Group

Mapping Class Group: History

Page 16: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Mapping Class Group: History

(Dehn)

Page 17: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Mapping Class Group: History

(Patel-Vlamis)

Page 18: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Mapping Class Group: History

(AF)

Page 19: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Classes of Handle Shifts

Page 20: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

One-Ended Handle Shifts

Page 21: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Handle Shifts - Relations

Page 22: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Basmajian, Ara. "Hyperbolic structures for surfaces of infinite type." Transactions of the American Mathematical Society 336.1 (1993): 421-444.

Farb, Benson, and Dan Margalit. A Primer on Mapping Class Groups (PMS-49). Princeton University Press, 2011.

Patel, Priyam, and Nicholas G. Vlamis. "Algebraic and topological properties of big mapping class groups." arXiv preprint arXiv:1703.02665 (2017).

Richards, Ian. "On the classification of noncompact surfaces." Transactions of the American Mathematical Society 106.2 (1963): 259-269.

Acknowledgements and CitationsThanks to:

Professor Dan Margalit Justin Lanier

Dr. Nick Vlamis

Page 23: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

Questions?

Infinite jail cell window

Page 24: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

What is the Loch Ness Monster?

Page 25: Mapping Class Groups of Infinite-Type Surfacespeople.math.gatech.edu/~dmargalit7/reu/BigMod_MathFest_2017.pdf · A Primer on Mapping Class Groups (PMS-49). Princeton University Press,

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