Geometric group theoryFrom Wikipedia, the free encyclopedia

Contents

1 (2,3,7) triangle group 11.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Hyperbolic construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Group presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Quaternion algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Relation to SL(2,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Amenable group 52.1 Denition for locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Equivalent conditions for amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Case of discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Artin group 113.1 Classes of Artin groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Artin groups of nite type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 Right-angled Artin groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Other Artin Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 BassSerre theory 144.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Basic set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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4.2.1 Graphs in the sense of Serre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.2 Graphs of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.3 Fundamental group of a graph of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.4 Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.5 The normal forms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 BassSerre covering trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Fundamental theorem of BassSerre theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.5.1 Amalgamated free product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5.2 HNN extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5.3 A graph with the trivial graph of groups structure . . . . . . . . . . . . . . . . . . . . . . 18

4.6 Basic facts and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.7 Trivial and nontrivial actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.8 Hyperbolic length functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.8.1 Basic facts regarding hyperbolic length functions . . . . . . . . . . . . . . . . . . . . . . . 194.8.2 Uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.9 Important developments in BassSerre theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.10 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Boundedly generated group 245.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4 Free groups are not boundedly generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.4.1 Burnside couterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4.2 Symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4.3 Hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4.4 Brooks pseudocharacters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4.5 Gromov boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Buekenhout geometry 286.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Building (mathematics) 307.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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7.4 Connection with BN pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.5 Spherical and ane buildings for SL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.5.1 Spherical building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.5.2 Ane building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.5.3 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.5.4 Geometric relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.5.5 Bruhat-Tits trees with complex multiplication . . . . . . . . . . . . . . . . . . . . . . . . 34

7.6 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Cayley graph 378.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.5 Schreier coset graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.6 Connection to group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.6.1 Geometric group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.7 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.8 Bethe lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9 Coxeter complex 449.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.1.1 The canonical linear representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.1.2 Chambers and the Tits cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.1.3 The Coxeter complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.2.1 Finite dihedral groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.2.2 The innite dihedral group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.3 Alternative construction of the Coxeter complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10 Dehn function 4710.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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10.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.2.1 Area of a relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.2.2 Isoperimetric function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.2.3 Dehn function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.2.4 Growth types of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.3 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.5 Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

11 Discrete group 5411.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

12 Flexagon 5712.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

12.1.1 Discovery and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.1.2 Attempted commercial development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

12.2 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.2.1 Tetraexagons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.2.2 Hexaexagons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2.3 Higher order exagons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

13 Free group 6413.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.4 Universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.5 Facts and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.6 Free abelian group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.7 Tarskis problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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13.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

14 Flner sequence 7014.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7014.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7014.3 Proof of amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

15 Geometric group action 7215.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

16 Geometric group theory 7316.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.2 Modern themes and developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

16.5.1 Books and monographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

17 Graph of groups 8217.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.2 Fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.3 Structure theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.5 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

18 Grigorchuk group 8418.1 History and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

18.2.1 Basic features of the Grigorchuk group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8518.3 Properties and facts regarding the Grigorchuk group[16] . . . . . . . . . . . . . . . . . . . . . . . . 8618.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

19 Gromov boundary 8819.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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19.1.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.2 Properties of the Gromov boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

19.4.1 Visual boundary of CAT(0) space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.5 Cannons Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

20 Gromovs theorem on groups of polynomial growth 9220.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

21 Grushko theorem 9421.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9421.2 History and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9421.3 Grushko decomposition theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9521.4 Sketch of the proof using Bass-Serre theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9521.5 Sketch of Stallings proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

21.5.1 Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9621.5.2 Construction of binding tie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9721.5.3 Proof of Grushko theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9721.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

22 Haagerup property 9922.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9922.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9922.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

23 Hanna Neumann conjecture 10123.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10123.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10123.3 Strengthened Hanna Neumann conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.4 Partial results and other generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

24 Hyperbolic group 10424.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

24.1.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10524.2 Examples of hyperbolic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10524.3 Examples of non-hyperbolic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10524.4 Homological characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10624.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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24.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10624.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10624.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10624.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

25 Iterated monodromy group 10825.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.2 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

25.3.1 Iterated monodromy groups of rational functions . . . . . . . . . . . . . . . . . . . . . . . 10925.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

26 Kazhdans property (T) 11026.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11026.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11026.3 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11126.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11126.5 Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11126.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11226.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

27 Kurosh subgroup theorem 11427.1 History and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.2 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.3 Proof using BassSerre theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.4 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

28 Length function 11628.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11628.2 Word metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11628.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

29 Out(Fn) 11729.1 Outer space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

29.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.1.2 Connection to length functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.1.3 Simplicial structure on the outer space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

29.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.3 Analogy with mapping class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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29.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

30 Presentation complex 11930.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

31 Quasi-isometry 12031.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12031.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12131.3 Equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12131.4 Use in geometric group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12131.5 Examples of quasi-isometry invariants of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

31.5.1 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12131.5.2 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12231.5.3 Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12231.5.4 Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12231.5.5 Asymptotic cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

31.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

32 Random group 12432.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12432.2 The few-relator model of random groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12432.3 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12432.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

33 Relatively hyperbolic group 12633.1 Intuitive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12633.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12633.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12633.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12633.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

34 Rips machine 12834.1 Actions of surface groups on R-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12834.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12834.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12834.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

35 Small cancellation theory 13035.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13035.2 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

35.2.1 Pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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35.2.2 Metric small cancellation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.2.3 Non-metric small cancellation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

35.3 Basic results of small cancellation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13235.3.1 Greendlingers lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13235.3.2 Dehns algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13235.3.3 Asphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13335.3.4 More general curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13335.3.5 Other basic properties of small cancellation groups . . . . . . . . . . . . . . . . . . . . . . 133

35.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13335.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.6 Basic references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13535.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13535.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

36 Stallings theorem about ends of groups 13836.1 Ends of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13836.2 Ends of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

36.2.1 Basic facts and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13836.2.2 Cuts and almost invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

36.3 Formal statement of Stallings theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13936.4 Applications and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14036.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14136.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

37 Tits alternative 14337.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

38 Train track map 14438.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

38.2.1 Combinatorial map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.2.2 Train track map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.2.3 Topological representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.2.4 Train track representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.2.5 Legal and illegal turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.2.6 Derivative map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

38.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14638.4 Main result for irreducible automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

38.4.1 Irreducible automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

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38.4.2 BestvinaHandel theorem for irreducible automorphisms . . . . . . . . . . . . . . . . . . 14638.5 Relative train tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.6 Applications and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14838.8 Basic references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14838.9 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14838.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

39 Triangle group 15039.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15039.2 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

39.2.1 The Euclidean case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.2.2 The spherical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.2.3 The hyperbolic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

39.3 von Dyck groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.4 Overlapping tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15339.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

40 Ultralimit 15440.1 Ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15440.2 Limit of a sequence of points with respect to an ultralter . . . . . . . . . . . . . . . . . . . . . . 15440.3 Ultralimit of metric spaces with specied base-points . . . . . . . . . . . . . . . . . . . . . . . . . 15440.4 On basepoints in the case of uniformly bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . 15540.5 Basic properties of ultralimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.6 Asymptotic cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15640.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15640.9 Basic References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15740.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

41 Van Kampen diagram 15841.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15841.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

41.2.1 Further terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15941.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15941.4 van Kampen lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

41.4.1 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16041.4.2 Strengthened version of van Kampens lemma . . . . . . . . . . . . . . . . . . . . . . . . 161

CONTENTS xi

41.5 Dehn functions and isoperimetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16141.5.1 Area of a word representing the identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 16141.5.2 Isoperimetric functions and Dehn functions . . . . . . . . . . . . . . . . . . . . . . . . . . 161

41.6 Generalizations and other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16241.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16241.8 Basic references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16241.9 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16341.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

42 Von Neumann conjecture 16442.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

43 Weyl distance function 16643.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16643.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16743.3 Abstract characterization of buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16743.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16743.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

44 Word metric 16844.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

44.1.1 The group of integers Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16844.1.2 The group Z Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

44.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.2.1 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

44.3 Example in a free group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.4 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

44.4.1 Isometry of the left action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.4.2 Bilipschitz invariants of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.4.3 Quasi-isometry invariants of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

44.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17144.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17144.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 172

44.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17444.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Chapter 1

(2,3,7) triangle group

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. Thisimportance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largestpossible order, 84(g 1), of its automorphism group.A note on terminology the "(2,3,7) triangle group most often refers, not to the full triangle group (2,3,7) (theCoxeter groupwith Schwarz triangle (2,3,7) or a realization as a hyperbolic reection group), but rather to the ordinarytriangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which is index 2.Torsion-free normal subgroups of the (2,3,7) triangle group are Fuchsian groups associated with Hurwitz surfaces,such as the Klein quartic, Macbeath surface and First Hurwitz triplet.

1.1 Constructions

1.1.1 Hyperbolic constructionTo construct the triangle group, start with a hyperbolic triangle with angles /2, /3, /7. This triangle, the smallesthyperbolic Schwarz triangle, tiles the plane by reections in its sides. Consider then the group generated by reectionsin the sides of the triangle, which (since the triangle tiles) is a non-Euclidean crystallographic group (discrete subgroupof hyperbolic isometries) with this triangle for fundamental domain; the associated tiling is the order-3 bisectedheptagonal tiling. The (2,3,7) triangle group is dened as the index 2 subgroup consisting of the orientation-preservingisometries, which is a Fuchsian group (orientation-preserving NEC group).

1.1.2 Group presentationIt has a presentation in terms of a pair of generators, g2, g3, modulo the following relations:

g22 = g33 = (g2g3)

7 = 1:

Geometrically, these correspond to rotations by 2/2, 2/3, 2/7 about the vertices of the Schwarz triangle.

1.1.3 Quaternion algebraThe (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable orderin a quaternion algebra. More specically, the triangle group is the quotient of the group of quaternions by its center1.Let = 2cos(2/7). Then from the identity

(2 )3 = 7( 1)2:

1

2 CHAPTER 1. (2,3,7) TRIANGLE GROUP

The (2,3,7) triangle group is the group of orientation-preserving isometries of the tiling by the (2,3,7) Schwarz triangle, shown herein a Poincar disk model projection.

we see that Q() is a totally real cubic extension of Q. The (2,3,7) hyperbolic triangle group is a subgroup of thegroup of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,jand relations i2 = j2 = , ij = ji. One chooses a suitable Hurwitz quaternion order QHur in the quaternion algebra.Here the order QHur is generated by elements

g2 =1 ij

g3 =12 (1 + (

2 2)j + (3 2)ij):

In fact, the order is a free Z[]-module over the basis 1; g2; g3; g2g3 . Here the generators satisfy the relations

g22 = g33 = (g2g3)

7 = 1;

which descend to the appropriate relations in the triangle group, after quotienting by the center.

1.2. RELATION TO SL(2,R) 3

Visualization of the map (2,3,) (2,3,7) by morphing the associated tilings.[1]

1.2 Relation to SL(2,R)

Extending the scalars fromQ() to R (via the standard imbedding), one obtains an isomorphism between the quater-nion algebra and the algebra M(2,R) of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibitthe (2,3,7) triangle group as a specic Fuchsian group in SL(2,R), specically as a quotient of the modular group.This can be visualized by the associated tilings, as depicted at right: the (2,3,7) tiling on the Poincar disc is a quotientof the modular tiling on the upper half-plane.However, for many purposes, explicit isomorphisms are unnecessary. Thus, traces of group elements (and hence alsotranslation lengths of hyperbolic elements acting in the upper half-plane, as well as systoles of Fuchsian subgroups)can be calculated by means of the reduced trace in the quaternion algebra, and the formula

tr() = 2 cosh(`/2):

1.3 References

[1] Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp

4 CHAPTER 1. (2,3,7) TRIANGLE GROUP

1.4 Further reading Elkies, N.: Shimura curve computations. Algorithmic number theory (Portland, OR, 1998), 147, LectureNotes in Computer Science, 1423, Springer, Berlin, 1998. See arXiv:math.NT/0005160

Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of systole of arithmetic Riemann surfaces along con-gruence subgroups. J. Dierential Geom. 76 (2007), no. 3, 399422. Available at arXiv:math.DG/0505007

Chapter 2

Amenable group

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operationon bounded functions that is invariant under translation by group elements. The original denition, in terms of anitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 underthe German name messbar (measurable in English) in response to the BanachTarski paradox. In 1949 MahlonM. Day introduced the English translation amenable, apparently as a pun.[1]

The amenability property has a large number of equivalent formulations. In the eld of analysis, the denition is interms of linear functionals. An intuitive way to understand this version is that the support of the regular representationis the whole space of irreducible representations.In discrete group theory, where G has the discrete topology, a simpler denition is used. In this setting, a group isamenable if one can say what proportion of G any given subset takes up.If a group has a Flner sequence then it is automatically amenable.

2.1 Denition for locally compact groupsLet G be a locally compact Hausdor group. Then it is well known that it possesses a unique, up-to-scale left- (orright-) rotation invariant ring measure, the Haar measure. (This is Borel regular measure whenG is second-countable;there are both left and right measures when G is compact.) Consider the Banach space L(G) of essentially boundedmeasurable functions within this measure space (which is clearly independent of the scale of the Haar measure).Denition 1. A linear functional in Hom(L(G), R) is said to be amean if has norm 1 and is non-negative, i.e.f 0 a.e. implies (f) 0.Denition 2. A mean in Hom(L(G), R) is said to be left-invariant (resp. right-invariant) if (gf) = (f) forall g in G, and f in L(G) with respect to the left (resp. right) shift action of gf(x) = f(g1x)(resp. fg(x) = f(xg1) ).Denition 3. A locally compact Hausdor group is called amenable if it admits a left- (or right-)invariant mean.

2.2 Equivalent conditions for amenabilityPier (1984) contains a comprehensive account of the conditions on a second countable locally compact group G thatare equivalent to amenability:[2]

Existence of a left (or right) invariant mean on L(G). The original denition, which depends on the axiomof choice.

Existence of left-invariant states. There is a left-invariant state on any separable left-invariant unital C*subalgebra of the bounded continuous functions on G.

Fixed-point property. Any action of the group by continuous ane transformations on a compact convexsubset of a (separable) locally convex topological vector space has a xed point. For locally compact abeliangroups, this property is satised as a result of the MarkovKakutani xed-point theorem.

5

6 CHAPTER 2. AMENABLE GROUP

Irreducible dual. All irreducible representations are weakly contained in the left regular representation onL2(G).

Trivial representation. The trivial representation of G is weakly contained in the left regular representation. Godement condition. Every bounded positive-denite measure on G satises (1) 0. Valette (1998) im-proved this criterion by showing that it is sucient to ask that, for every continuous positive-denite compactlysupported function f on G, the function f has non-negative integral with respect to Haar measure, where denotes the modular function.

Days asymptotic invariance condition. There is a sequence of integrable non-negative functions n withintegral 1 on G such that (g)n n tends to 0 in the weak topology on L1(G).

Reiters condition. For every nite (or compact) subset F of G there is an integrable non-negative function with integral 1 such that (g) is arbitrarily small in L1(G) for g in F.

Dixmiers condition. For every nite (or compact) subset F of G there is unit vector f in L2(G) such that(g)f f is arbitrarily small in L2(G) for g in F.

GlicksbergReiter condition. For any f in L1(G), the distance between 0 and the closed convex hull in L1(G)of the left translates (g)f equals |f |.

Flner condition. For every nite (or compact) subset F of G there is a measurable subset U of G with nitepositive Haar measure such that m(U gU)/m(U) is arbitrarily small for g in F.

Leptins condition. For every nite (or compact) subset F ofG there is a measurable subsetU ofG with nitepositive Haar measure such that m(FU U)/m(U) is arbitrarily small.

Kestens condition. Left convolution on L2(G) by a symmetric probability measure on G gives an operator ofoperator norm 1.

Johnsons cohomological condition. The Banach algebra A = L1(G) is amenable as a Banach algebra, i.e.any bounded derivation of A into the dual of a Banach A-bimodule is inner.

2.3 Case of discrete groupsThe denition of amenability is simpler in the case of a discrete group,[3] i.e. a group equipped with the discretetopology.[4]

Denition. A discrete group G is amenable if there is a nitely additive measure (also called a mean) a functionthat assigns to each subset of G a number from 0 to 1such that

1. The measure is a probability measure: the measure of the whole group G is 1.2. The measure is nitely additive: given nitely many disjoint subsets of G, the measure of the union of the sets

is the sum of the measures.3. The measure is left-invariant: given a subset A and an element g of G, the measure of A equals the measure

of gA. (gA denotes the set of elements ga for each element a in A. That is, each element of A is translated onthe left by g.)

This denition can be summarized thus: G is amenable if it has a nitely-additive left-invariant probability measure.Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a randomelement of G is in A?It is a fact that this denition is equivalent to the denition in terms of L(G).Having a measure on G allows us to dene integration of bounded functions on G. Given a bounded function f : G R, the integral

ZG

f d

2.4. PROPERTIES 7

is dened as in Lebesgue integration. (Note that some of the properties of the Lebesgue integral fail here, since ourmeasure is only nitely additive.)If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure , thefunction (A) = (A1) is a right-invariant measure. Combining these two gives a bi-invariant measure:

(A) =

Zg2G

Ag1

d:

The equivalent conditions for amenability also become simpler in the case of a countable discrete group . For sucha group the following conditions are equivalent:[5]

is amenable.

If acts by isometries on a (separable) Banach space E, leaving a weakly closed convex subset C of the closedunit ball of E* invariant, then has a xed point in C.

There is a left invariant norm-continuous functional on () with (1) = 1 (this requires the axiom ofchoice).

There is a left invariant state on any left invariant separable unital C* subalgebra of ().

There is a set of probability measures n on such that ||g n n||1 tends to 0 for each g in (M.M. Day).

There are unit vectors xn in 2() such that ||g xn xn||2 tends to 0 for each g in (J. Dixmier).

There are nite subsets Sn of such that |g Sn Sn| / |Sn| tends to 0 for each g in (Flner).

If is a symmetric probability measure on with support generating , then convolution by denes anoperator of norm 1 on 2() (Kesten).

If acts by isometries on a (separable) Banach space E and f in (, E*) is a bounded 1-cocycle, i.e. f(gh)= f(g) + gf(h), then f is a 1-coboundary, i.e. f(g) = g for some in E* (B.E. Johnson).

The von Neumann group algebra of is hypernite (A. Connes).

Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group ishypernite, so the last condition no longer applies in the case of connected groups.Amenability is related to the spectral problem of Laplacians. For instance, the fundamental group of a closed Rie-mannian manifold is amenable if and only if the bottom of the spectrum of the Laplacian is 0 (R. Brooks, T. Sunada).

2.4 Properties Every (closed) subgroup of an amenable group is amenable.

Every quotient of an amenable group is amenable.

A group extension of an amenable group by an amenable group is again amenable. In particular, nite directproduct of amenable groups are amenable, although innite products need not be.

Direct limits of amenable groups are amenable. In particular, if a group can be written as a directed union ofamenable subgroups, then it is amenable.

Amenable groups are unitarizable; the converse is an open problem.

Countable discrete amenable groups obey the Ornstein isomorphism theorem.[6][7]

8 CHAPTER 2. AMENABLE GROUP

2.5 Examples

Finite groups are amenable. Use the counting measure with the discrete denition. More generally, compactgroups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).

The group of integers is amenable (a sequence of intervals of length tending to innity is a Flner sequence).The existence of a shift-invariant, nitely additive probability measure on the group Z also follows easily fromthe HahnBanach theorem this way. Let S be the shift operator on the sequence space (Z), which is denedby (Sx)i = xi for all x (Z), and let u (Z) be the constant sequence ui = 1 for all i Z. Any element y Y:=Ran(S I) has a distance larger than or equal to 1 from u (otherwise yi = xi+1 - xi would be positive andbounded away from zero, whence xi could not be bounded). This implies that there is a well-dened norm-onelinear form on the subspace Ru + Y taking tu + y to t. By the HahnBanach theorem the latter admits a norm-one linear extension on (Z), which is by construction a shift-invariant nitely additive probability measureon Z.

If every conjugacy class in a locally compact group has compact closure, then the group is amenable. Examplesof groups with this property include compact groups, locally compact abelian groups, and discrete groups withnite conjugacy classes.[8]

By the direct limit property above, a group is amenable if all its nitely generated subgroups are. That is, locallyamenable groups are amenable.

By the fundamental theorem of nitely generated abelian groups, it follows that abelian groups areamenable.

It follows from the extension property above that a group is amenable if it has a nite index amenable subgroup.That is, virtually amenable groups are amenable.

Furthermore, it follows that all solvable groups are amenable.

All examples above are elementary amenable. The rst class of examples below can be used to exhibit non-elementaryamenable examples thanks to the existence of groups of intermediate growth.

Finitely generated groups of subexponential growth are amenable. A suitable subsequence of balls will providea Flner sequence.[9]

Finitely generated innite simple groups cannot be obtained by bootstrap constructions as used to constructelementary amenable groups. Since there exist such simple groups that are amenable, due to Juschenko andMonod,[10] this provides again non-elementary amenable examples.

2.6 CounterexamplesIf a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable. Theconverse to this statement is the so-called von Neumann conjecture, which was disproved by Olshanskii in 1980 usinghis Tarski monsters. Adyan subsequently showed that free Burnside groups are non-amenable: since they are periodic,they cannot contain the free group on two generators. These groups are nitely generated, but not nitely presented.However, in 2002 Sapir and Olshanskii found nitely presented counterexamples: non-amenable nitely presentedgroups that have a periodic normal subgroup with quotient the integers.[11]

For nitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative:[12] everysubgroup of GL(n,k) with k a eld either has a normal solvable subgroup of nite index (and therefore is amenable)or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later foundan analytic proof based on V. Oseledets' multiplicative ergodic theorem.[13] Analogues of the Tits alternative havebeen proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes ofnon-positive curvature.[14]

2.7. SEE ALSO 9

2.7 See also

Uniformly bounded representation

Kazhdans property (T)

Von Neumann conjecture

2.8 Notes

[1] Days rst published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups andgroups, Bull. A.M.S. 55 (1949) 10541055. Many text books on amenabilty, such as Volker Rundes, suggest that Daychose the word as a pun.

[2] Pier 1984

[3] See:

Greenleaf 1969 Pier 1984 Takesaki 2002a Takesaki 2002b

[4] Weisstein, Eric W., Discrete Group, MathWorld.

[5] Pier 1984

[6] Ornstein, D.; Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math.48: 1141.

[7] Lewis Bowen (2011), "Every countably innite group is almost Ornstein", ArXiv abs/1103.4424

[8] Leptin 1968

[9] See:

Greenleaf 1969 Pier 1984 Takesaki 2002a Takesaki 2002b

[10] Juschenko, Kate; Monod, Nicolas (2013), Cantor systems, piecewise translations and simple amenable groups, Annalsof Mathematics 178 (2): 775787, doi:10.4007/annals.2013.178.2.7

[11] Olshanskii, Alexander Yu.; Sapir, MarkV. (2002), Non-amenable nitely presented torsion-by-cyclic groups, Publ. Math.Inst. Hautes tudes Sci. 96: 43169, doi:10.1007/s10240-002-0006-7

[12] Tits, J. (1972), Free subgroups in linear groups, J. Algebra 20 (2): 250270, doi:10.1016/0021-8693(72)90058-0

[13] Guivarc'h, Yves (1990), Produits de matrices alatoires et applications aux proprits gometriques des sous-groupes dugroupes linaire, Ergod. Th. & Dynam. Sys. 10 (3): 483512, doi:10.1017/S0143385700005708

[14] Ballmann, Werner; Brin, Michael (1995), Orbihedra of nonpositive curvature, Inst. Hautes tudes Sci. Publ. Math. 82:169209, doi:10.1007/BF02698640

10 CHAPTER 2. AMENABLE GROUP

2.9 ReferencesThis article incorporates material from Amenable group on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Brooks, Robert (1981), The fundamental group and the spectrum of the laplacian, Comment. Math. Helv.56: 581598, doi:10.1007/bf02566228

Dixmier, Jacques (1977), C*-algebras (translated from the French by Francis Jellett), North-Holland Mathe-matical Library 15, North-Holland

Greenleaf, F.P. (1969), InvariantMeans on Topological Groups and Their Applications, VanNostrand Reinhold Juschenko, Kate; Monod, Nicolas (2013), Cantor systems, piecewise translations and simple amenable groups,Annals of Mathematics 178 (2): 775787, doi:10.4007/annals.2013.178.2.7

Leptin, H. (1968), Zur harmonischen Analyse klassenkompakter Gruppen, Invent. Math. 5: 249254,doi:10.1007/bf01389775

Pier, Jean-Paul (1984),Amenable locally compact groups, Pure andAppliedMathematics,Wiley, Zbl 0621.43001 Runde, V. (2002), Lectures on Amenability, LectureNotes inMathematics 1774, Springer, ISBN9783540428527 Sunada, Toshikazu (1989), Unitary representations of fundamental groups and the spectrum of twisted Lapla-cians, Topology 28: 125132, doi:10.1016/0040-9383(89)90015-3

Takesaki, M. (2002a), Theory of Operator Algebras 2, Springer, ISBN 9783540422488 Takesaki, M. (2002b), Theory of Operator Algebras 3, Springer, ISBN 9783540429142

Valette, Alain (1998), On Godements characterisation of amenability, Bull. Austral. Math. Soc. 57: 153158, doi:10.1017/s0004972700031506

von Neumann, J (1929), Zur allgemeinen Theorie des Maes (PDF), Fund. Math. 13 (1): 73111

2.10 External links Some notes on amenability by Terry Tao

Chapter 3

Artin group

In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form

Dx1; x2; : : : ; xn

hx1; x2im1;2 = hx2; x1im2;1 ; : : : ; hxn1; xnimn1;n = hxn; xn1imn;n1Ewhere

mi;j = mj;i 2 f2; 3; : : : ;1g

Form 1: Severalimportant classes of Artin groups can be dened in terms of the properties of the Coxeter matrix.

3.1.1 Artin groups of nite type

If M is a Coxeter matrix of nite type, so that the corresponding Coxeter group W = A(M) is nite, then the Artingroup A = A(M) is called an Artin group of nite type. The 'irreducible types are labeled as An , Bn = Cn , Dn ,I2(n) , F4 , E6 , E7 , E8 , H3 , H4 . A pure Artin group of nite type can be realized as the fundamental group of thecomplement of a nite hyperplane arrangement in Cn. Pierre Deligne and Brieskorn-Saito have used this geometricdescription to compute the center of A, its cohomology, and to solve the word and conjugacy problems.

11

12 CHAPTER 3. ARTIN GROUP

3.1.2 Right-angled Artin groupsIfM is a matrix all of whose elements are equal to 2 or , then the corresponding Artin group is called a right-angledArtin group, but also a (free) partially commutative group, graph group, trace group, semifree group or evenlocally free group. For this class of Artin groups, a dierent labeling scheme is commonly used. Any graph onn vertices labeled 1, 2, , n denes a matrix M, for which mij = 2 if i and j are connected by an edge in , and mij= otherwise. The right-angled Artin group A() associated with the matrix M has n generators x1, x2, , x andrelations

xixj = xjxi whenever i and j are connected by an edge in :

The class of right-angled Artin groups includes the free groups of nite rank, corresponding to a graph with no edges,and the nitely-generated free abelian groups, corresponding to a complete graph. In fact, every right-angled Artingroup of rank r can be constructed as HNN extension of a right-angled Artin group of rank r1, with the free productand direct product as the extreme cases. A generalization of this construction is called a graph product of groups.A right-angled Artin group is a special case of this product, with every vertex/operand of the graph-product being afree group of rank one (the innite cyclic group).Mladen Bestvina and Noel Brady constructed a nonpositively curved cubical complex K whose fundamental groupis a given right-angled Artin group A(). They applied Morse-theoretic arguments to their geometric description ofArtin groups and exhibited rst known examples of groups with the property (FP2) that are not nitely presented.

3.2 Other Artin GroupsWe dene that an Artin group or a Coxeter group is of large type if m 3 for all i j. We say that an Artin groupor a Coxeter group is of extra-large type if m 4 for all i j.Kenneth Appel and P.E. Schupp looked further into Artin groups and the properties that hold true for them. Theyproved four theorems, which were known to be true for Coxeter groups, and showed that they also held for Artingroups. Appel and Schupp had discovered that they could study extra-large Artin and Coxeter groups through thetechniques of small cancellation theory. They also discovered that they could use a renement of these sametechniques to work with these groups of large type.[1]

Theorem 1 : Let G be an Artin or Coxeter group of extra-large type. If J I then GJ has a presentation dened bythe Coxeter matrix MJ and the generalized word problem for GJ in G is solvable. If J, K I then GJ GK = GJ K.Theorem 2 : An Artin group of extra-large type is torsion-free.Theorem 3 : Let G be an Artin group of extra-large type. Then the set {a2 : i I} freely generates a free subgroupof G.Theorem 4 : An Artin or Coxeter group of extra-large type has solvable conjugacy problem.

3.3 See also Free partially commutative monoid Artinian group (an unrelated notion) Non-commutative cryptography Elementary abelian group

3.4 Notes[1] Appel, Kenneth I.; Schupp, P. E. (1983), Artin Groups and Innite Coxeter Groups, Inventiones Mathematicae 72 (2):

201220, doi:10.1007/BF01389320

3.5. REFERENCES 13

3.5 References Bestvina, Mladen; Brady, Noel (1997), Morse theory and niteness properties of groups, Invent. Math. 129(3): 445470, doi:10.1007/s002220050168

Deligne, Pierre (1972), Les immeubles des groupes de tresses gnraliss, Invent. Math. 17 (4): 273302,doi:10.1007/BF01406235

Brieskorn, Egbert; Saito, Kyoji (1972), Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (4): 245271, doi:10.1007/bf01406235

Charney, Ruth (October 2006), An Introduction to Right-Angled Artin Groups (PDF), Department of Mathe-matics, Brandeis University

Casals-Ruiz, Montserrat; Kazachkov, Ilya (2009), On Systems of Equations over Free Partially CommutativeGroups, arXiv:0810.4867

Esyp, E. S.; Kazachkov, I. V.; Remeslennikov, V. N. (2005), Divisibility Theory and Complexity of Algorithmsin Free Partially Commutative Groups, arXiv:math/0512401

Hermiller, Susan; Meier, John, Algorithms and geometry for graph products of groups (PDF)

Chapter 4

BassSerre theory

BassSerre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraicstructure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees withdecomposing groups as iterated applications of the operations of free product with amalgamation and HNN ex-tension, via the notion of the fundamental group of a graph of groups. BassSerre theory can be regarded asone-dimensional version of the orbifold theory.

4.1 HistoryBassSerre theory was developed by Jean-Pierre Serre in the 1970s and formalized in Trees, Serres seminal 1977monograph (developed in collaboration with Hyman Bass) on the subject.[1][2] Serres original motivation was to un-derstand the structure of certain algebraic groups whose BruhatTits buildings are trees. However, the theory quicklybecame a standard tool of geometric group theory and geometric topology, particularly the study of 3-manifolds. Sub-sequent work of Hyman Bass[3] contributed substantially to the formalization and development of basic tools of thetheory and currently the term BassSerre theory is widely used to describe the subject.Mathematically, BassSerre theory builds on exploiting and generalizing the properties of two older group-theoreticconstructions: free product with amalgamation and HNN extension. However, unlike the traditional algebraic study ofthese two constructions, BassSerre theory uses the geometric language of covering theory and fundamental groups.Graphs of groups, which are the basic objects of BassSerre theory, can be viewed as one-dimensional versions oforbifolds.Apart from Serres book,[2] the basic treatment of BassSerre theory is available in the article of Bass,[3] the articleof Scott and Wall[4] and the books of Hatcher,[5] Baumslag,[6] Dicks and Dunwoody[7] and Cohen.[8]

4.2 Basic set-up

4.2.1 Graphs in the sense of Serre

Serres formalism of graphs is slightly dierent from the standard formalism from graph theory. Here a graph Aconsists of a vertex set V, an edge set E, an edge reversal map E ! E; e 7! e such that e e and e = e for everye in E, and an initial vertex map o : E V. Thus in A every edge e comes equipped with its formal inverse e. Thevertex o(e) is called the origin or the initial vertex of e and the vertex o(e) is called the terminus of e and is denotedt(e). Both loop-edges (that is, edges e such that o(e) = t(e)) and multiple edges are allowed. An orientation on A is apartition of E into the union of two disjoint subsets E+ and E so that for every edge e exactly one of the edges fromthe pair e, e belongs to E+ and the other belongs to E.

4.2.2 Graphs of groups

A graph of groups A consists of the following data:

14

4.2. BASIC SET-UP 15

A connected graph A; An assignment of a vertex group Av to every vertex v of A. An assignment of an edge group Ae to every edge e of A so that we have Ae = Ae for every e E. Boundary monomorphisms e : Ae ! Ao(e) for all edges e of A, so that each e is an injective grouphomomorphism.

For every eE the map e : Ae ! At(e) is also denoted by e.

4.2.3 Fundamental group of a graph of groupsThere are two equivalent denitions of the notion of the fundamental group of a graph of groups: the rst is a directalgebraic denition via an explicit group presentation (as a certain iterated application of amalgamated free productsand HNN extensions), and the second using the language of groupoids.The algebraic denition is easier to state:First, choose a spanning tree T in A. The fundamental group of A with respect to T, denoted 1(A, T), is dened asthe quotient of the free product

(v2VAv) F (E)where F(E) is a free group with free basis E, subject to the following relations:

ee(g)e = e(g) for every e in E and every g 2 Ae . (The so-called BassSerre relation.) ee = 1 for every e in E. e = 1 for every edge e of the spanning tree T.

There is also a notion of the fundamental group of A with respect to a base-vertex v in V, denoted 1(A, v), whichis dened using the formalism of groupoids. It turns out that for every choice of a base-vertex v and every spanningtree T in A the groups 1(A, T) and 1(A, v) are naturally isomorphic.The fundamental group of a graph of groups has a natural topological interpretation as well: it is the fundamentalgroup of a graph of spaces whose vertex spaces and edge spaces have the fundamental groups of the vertex groupsand edge groups, respectively, and whose gluing maps induce the homomorphisms of the edge groups into the vertexgroups. One can therefore take this as a third denition of the fundamental group of a graph of groups.

Fundamental groups of graphs of groups as iterations of amalgamated products and HNN-extensions

The groupG = 1(A, T) dened above admits an algebraic description in terms of iterated amalgamated free productsand HNN extensions. First, form a group B as a quotient of the free product

(v2VAv) F (E+T )subject to the relations

e1e(g)e = e(g) for every e in E+T and every g 2 Ae . e = 1 for every e in E+T.

This presentation can be rewritten as

B = v2VAv/nclfe(g) = !e(g); where e 2 E+T; g 2 Geg

16 CHAPTER 4. BASSSERRE THEORY

which shows that B is an iterated amalgamated free product of the vertex groups Av.Then the group G = 1(A, T) has the presentation

hB;E+(A T )je1e(g)e = !e(g) where e 2 E+(A T ); g 2 Gei;which shows that G = 1(A, T) is a multiple HNN extension of B with stable letters feje 2 E+(A T )g .

4.2.4 SplittingsAn isomorphism between a group G and the fundamental group of a graph of groups is called a splitting of G. If theedge groups in the splitting come from a particular class of groups (e.g. nite, cyclic, abelian, etc.), the splitting issaid to be a splitting over that class. Thus a splitting where all edge groups are nite is called a splitting over nitegroups.Algebraically, a splitting of G with trivial edge groups corresponds to a free product decomposition

G = (Av) F (X)where F(X) is a free group with free basis X = E+(AT) consisting of all positively oriented edges (with respect tosome orientation on A) in the complement of some spanning tree T of A.

4.2.5 The normal forms theoremLet g be an element of G = 1(A, T) represented as a product of the form

g = a0e1a1 : : : enan;

where e1, ..., en is a closed edge-path in A with the vertex sequence v0, v1, ..., vn = v0 (that is v0=o(e1), vn = t(en)and vi = t(ei) = o(ei) for 0 < i < n) and where ai 2 Avi for i = 0, ..., n.Suppose that g = 1 in G. Then

either n = 0 and a0 = 1 in Av0 , or n > 0 and there is some 0 < i < n such that ei = ei and ai 2 !ei(Aei) .

The normal forms theorem immediately implies that the canonical homomorphisms Av 1(A, T) are injective, sothat we can think of the vertex groups Av as subgroups of G.Higgins has given a nice version of the normal form using the fundamental groupoid of a graph of groups.[9] Thisavoids choosing a base point or tree, and has been exploited in.[10]

4.3 BassSerre covering treesTo every graph of groups A, with a specied choice of a base-vertex, one can associate a BassSerre covering tree~A , which is a tree that comes equipped with a natural group action of the fundamental group 1(A, v) withoutedge-inversions. Moreover, the quotient graph ~A/1(A; v) is isomorphic to A.Similarly, if G is a group acting on a tree X without edge-inversions (that is, so that for every edge e of X and every gin G we have ge e), one can dene the natural notion of a quotient graph of groups A. The underlying graph A of Ais the quotient graph X/G. The vertex groups of A are isomorphic to vertex stabilizers in G of vertices of X and theedge groups of A are isomorphic to edge stabilizers in G of edges of X.Moreover, if X was the BassSerre covering tree of a graph of groups A and if G = 1(A, v) then the quotient graphof groups for the action of G on X can be chosen to be naturally isomorphic to A.

4.4. FUNDAMENTAL THEOREM OF BASSSERRE THEORY 17

4.4 Fundamental theorem of BassSerre theoryLet G be a group acting on a tree X without inversions. Let A be the quotient graph of groups and let v be a base-vertex in A. Then G is isomorphic to the group 1(A, v) and there is an equivariant isomorphism between the treeX and the BassSerre covering tree ~A . More precisely, there is a group isomorphism : G 1(A, v) and a graphisomorphism j : X ! ~A such that for every g in G, for every vertex x of X and for every edge e of X we have j(gx)= g j(x) and j(ge) = g j(e).One of the immediate consequences of the above result is the classic Kurosh subgroup theorem describing the alge-braic structure of subgroups of free products.

4.5 Examples

4.5.1 Amalgamated free productConsider a graph of groups A consisting of a single non-loop edge e (together with its formal inverse e) with twodistinct end-vertices u = o(e) and v = t(e), vertex groups H = Au, K = Av, an edge group C = Ae and the boundarymonomorphisms = e : C ! H;! = !e : C ! K . Then T = A is a spanning tree in A and the fundamentalgroup 1(A, T) is isomorphic to the amalgamated free product

G = H C K = H K/nclf(c) = !(c); c 2 Cg:In this case the BassSerre treeX = ~A can be described as follows. The vertex set of X is the set of cosets

V X = fgK : g 2 Gg t fgH : g 2 Gg:Two vertices gK and fH are adjacent in X whenever there exists k K such that fH = gkH (or, equivalently, wheneverthere is h H such that gK = fhK).The G-stabilizer of every vertex of X of type gK is equal to gKg1 and the G-stabilizer of every vertex of X of typegH is equal to gHg1. For an edge [gH, ghK] of X its G-stabilizer is equal to gh(C)h1g1.For every c C and h 'k K' the edges [gH, ghK] and [gH, gh(c)K] are equal and the degree of the vertex gH inX is equal to the index [H:(C)]. Similarly, every vertex of type gK has degree [K:(C)] in X.

4.5.2 HNN extensionLet A be a graph of groups consisting of a single loop-edge e (together with its formal inverse e), a single vertex v =o(e) = t(e), a vertex group B = Av, an edge group C = Ae and the boundary monomorphisms = e : C ! B;! =!e : C ! B . Then T = v is a spanning tree in A and the fundamental group 1(A, T) is isomorphic to the HNNextension

G = hB; eje1(c)e = !(c); c 2 Ci:with the base group B, stable letter e and the associated subgroups H = (C), K = (C) in B. The composition = ! 1 : H ! K is an isomorphism and the above HNN-extension presentation of G can be rewritten as

G = hB; eje1he = (h); h 2 Hi:In this case the BassSerre tree X = ~A can be described as follows. The vertex set of X is the set of cosets VX ={gB : g G}.Two vertices gB and fB are adjacent in X whenever there exists b in B such that either fB = gbeB or fB = gbe1B. TheG-stabilizer of every vertex of X is conjugate to B in G and the stabilizer of every edge of X is conjugate to H in G.Every vertex of X has degree equal to [B : H] + [B : K].

18 CHAPTER 4. BASSSERRE THEORY

4.5.3 A graph with the trivial graph of groups structureLet A be a graph of groups with underlying graph A such that all the vertex and edge groups in A are trivial. Let v bea base-vertex in A. Then 1(A,v) is equal to the fundamental group 1(A,v) of the underlying graph A in the standardsense of algebraic topology and the BassSerre covering tree ~A is equal to the standard universal covering space ~Aof A. Moreover, the action of 1(A,v) on ~A is exactly the standard action of 1(A,v) on ~A by deck transformations.

4.6 Basic facts and properties IfA is a graph of groups with a spanning tree T and if G = 1(A, T), then for every vertex v of A the canonicalhomomorphism from Av to G is injective.

If g G is an element of nite order then g is conjugate in G to an element of nite order in some vertex groupAv.

If F G is a nite subgroup then F is conjugate in G to a subgroup of some vertex group Av. If the graph A is nite and all vertex groups Av are nite then the group G is virtually free, that is, G containsa free subgroup of nite index.

If A is nite and all the vertex groups Av are nitely generated then G is nitely generated. If A is nite and all the vertex groups Av are nitely presented and all the edge groups Ae are nitely generatedthen G is nitely presented.

4.7 Trivial and nontrivial actionsA graph of groups A is called trivial if A = T is already a tree and there is some vertex v of A such that Av = 1(A,A). This is equivalent to the condition that A is a tree and that for every edge e = [u, z] of A (with o(e) = u, t(e) = z)such that u is closer to v than z we have [Az : e(Ae)] = 1, that is Az = e(Ae).An action of a group G on a tree X without edge-inversions is called trivial if there exists a vertex x of X that is xedby G, that is such that Gx = x. It is known that an action of G on X is trivial if and only if the quotient graph of groupsfor that action is trivial.Typically, only nontrivial actions on trees are studied in BassSerre theory since trivial graphs of groups do notcarry any interesting algebraic information, although trivial actions in the above sense (e. g. actions of groups byautomorphisms on rooted trees) may also be interesting for other mathematical reasons.One of the classic and still important results of the theory is a theorem of Stallings about ends of groups. The theoremstates that a nitely generated group has more than one end if and only if this group admits a nontrivial splitting overnite subroups that is, if and only if the group admits a nontrivial action without inversions on a tree with nite edgestabilizers.[11]

An important general result of the theory states that ifG is a group with Kazhdans property (T) thenG does not admitany nontrivial splitting, that is, that any action of G on a tree X without edge-inversions has a global xed vertex.[12]

4.8 Hyperbolic length functionsLet G be a group acting on a tree X without edge-inversions.For every gG put

`X(g) = minfd(x; gx)jx 2 V Xg:

Then X(g) is called the translation length of g on X.The function

4.9. IMPORTANT DEVELOPMENTS IN BASSSERRE THEORY 19

`X : G! Z; g 2 G 7! `X(g)

is called the hyperbolic length function or the translation length function for the action of G on X.

4.8.1 Basic facts regarding hyperbolic length functions

For g G exactly one of the following holds:

(a) X(g) = 0 and g xes a vertex of G. In this case g is called an elliptic element of G.(b) X(g) > 0 and there is a unique bi-innite embedded line in X, called the axis of g and denoted Lgwhich is g-invariant. In this case g acts on Lg by translation of magnitude X(g) and the element g Gis called hyperbolic.

If X(G) 0 then there exists a unique minimal G-invariant subtree XG of X. Moreover XG is equal to theunion of axes of hyperbolic elements of G.

The length-function X : G Z is said to be abelian if it is a group homomorphism from G to Z and non-abelianotherwise. Similarly, the action of G on X is said to be abelian if the associated hyperbolic length function is abelianand is said to be non-abelian otherwise.In general, an action of G on a tree X without edge-inversions is said to be minimal if there are no proper G-invariantsubtrees in X.An important fact in the theory says that minimal non-abelian tree actions are uniquely determined by their hyperboliclength functions:[13]

4.8.2 Uniqueness theorem

Let G be a group with two nonabelian minimal actions without edge-inversions on trees X and Y. Suppose that thehyperbolic length functions X and Y on G are equal, that is X(g) = Y(g) for every g G. Then the actions of Gon X and Y are equal in the sense that there exists a graph isomorphism f : X Y which is G-equivariant, that isf(gx) = g f(x) for every g G and every x VX.

4.9 Important developments in BassSerre theoryImportant developments in BassSerre theory in the last 30 years include:

Various accessibility results for nitely presented groups that bound the complexity (that is, the number ofedges) in a graph of groups decomposition of a nitely presented group, where some algebraic or geometricrestrictions on the types of groups considered are imposed. These results include:

Dunwoodys theorem about accessibility of nitely presented groups[14] stating that for any nitely pre-sented group G there exists a bound on the complexity of splittings of G over nite subgroups (the split-tings are required to satisfy a technical assumption of being reduced);

BestvinaFeighn generalized accessibility theorem[15] stating that for any nitely presented group G thereis a bound on the complexity of reduced splittings of G over small subgroups (the class of small groupsincludes, in particular, all groups that do not contain non-abelian free subgroups);

Acylindrical accessibility results for nitely presented (Sela,[16] Delzant[17]) and nitely generated (Weidmann[18])groups which bound the complexity of the so-called acylindrical splittings, that is splittings where fortheir BassSerre covering trees the diameters of xed subsets of nontrivial elements of G are uniformlybounded.

20 CHAPTER 4. BASSSERRE THEORY

The theory of JSJ-decompositions for nitely presented groups. This theory was motivated by the classic notionof JSJ decomposition in 3-manifold topology and was initiated, in the context of word-hyperbolic groups, bythe work of Sela. JSJ decompositions are splittings of nitely presented groups over some classes of smallsubgroups (cyclic, abelian, noetherian, etc., depending on the version of the theory) that provide a canonicaldescriptions, in terms of some standard moves, of all splittings of the group over subgroups of the class. Thereare a number of versions of JSJ-decomposition theories:

The initial version of Sela for cyclic splittings of torsion-free word-hyperbolic groups.[19] Bowditchs version of JSJ theory for word-hyperbolic groups (with possible torsion) encoding their split-tings over virtually cyclic subgroups.[20]

The version of Rips and Sela of JSJ decompositions of torsion-free nitely presented groups encodingtheir splittings over free abelian subgroups.[21]

The version of Dunwoody and Sageev of JSJ decompositions of nitely presented groups over noetheriansubgroups.[22]

The version of Fujiwara and Papasoglu, also of JSJ decompositions of nitely presented groups overnoetherian subgroups.[23]

A version of JSJ decomposition theory for nitely presented groups developed by Scott and Swarup.[24]

The theory of lattices in automorphism groups of trees. The theory of tree lattices was developed by Bass,Kulkarni and Lubotzky[25][26] by analogy with the theory of lattices in Lie groups (that is discrete subgroups ofLie groups of nite co-volume). For a discrete subgroup G of the automorphism group of a locally nite treeX one can dene a natural notion of volume for the quotient graph of groups A as

vol(A) =Xv2V

1

jAvj :

The group G is called an X-lattice if vol(A)< . The theory of tree lattices turns out to be useful in thestudy of discrete subgroups of algebraic groups over non-archimedean local elds and in the study ofKacMoody groups.

Development of foldings and Nielsen methods for approximating group actions on trees and analyzing theirsubgroup structure.[15][18][27][28]

The theory of ends and relative ends of groups, particularly various generalizations of Stallings theorem aboutgroups with more than one end.[29][30][31]

Quasi-isometric rigidity results for groups acting on trees.[32]

4.10 GeneralizationsThere have been several generalizations of BassSerre theory:

The theory of complexes of groups (see Haeiger,[33] Corson[34] Bridson-Haeiger[35]) provides a higher-dimensional generalization of BassSerre theory. The notion of a graph of groups is replaced by that of acomplex of groups, where groups are assigned to each cell in a simplicial complex, together with monomor-phisms between these groups corresponding to face inclusions (these monomorphisms are required to satisfycertain compatibility conditions). One can then dene an analog of the fundamental group of a graph of groupsfor a complex of groups. However, in order for this notion to have good algebraic properties (such as embed-dability of the vertex groups in it) and in order for a good analog for the notion of the BassSerre covering treeto exist in this context, one needs to require some sort of non-positive curvature condition for the complexof groups in question (see, for example [36][37]).

4.11. SEE ALSO 21

The theory of isometric group actions on real trees (or R-trees) which are metric spaces generalizing thegraph-theoretic notion of a tree (graph theory). The theory was developed largely in the 1990s, where theRips machine of Eliyahu Rips on the structure theory of stable group actions on R-trees played a key role(see Bestvina-Feighn[38]). This structure theory assigns to a stable isometric action of a nitely generatedgroup G a certain normal form approximation of that action by a stable action of G on a simplicial treeand hence a splitting of G in the sense of BassSerre theory. Group actions on real trees arise naturally inseveral contexts in geometric topology: for example as boundary points of the