601B -- Introduction to Geometric Group Theory (Fall 2023)

Class schedule and suggested exercises

Week 15: Residual finiteness; student presentations on Dehn functions and amenability

Week 14: Student presentations on asymptotic dimension, the Grigorchuk group, and free groups and foldings

Week 13: Guest lectures on Thompson's group F and hyperbolicity of the curve graph via unicorn paths

Week 12: Abstract commensurator groups; student presentations on the dynamical criterion for quasi-isometry

Week 11: Ends of groups, Stallings' theorem, Bass--Serre theory

Suggested reading: Löh 8.1-8.2, Scott and Wall's article "Topological methods in group theory" (see also Serre's book "Trees")

Suggested exercises: Löh 8.E.1, 8.E.6, 8.E.7

Week 10: Nonelementary hyperbolic groups have free subgroups; Mostow rigidity

Suggested reading: Löh 8.4, Kapovich and Benakli's survey article "Boundaries of hyperbolic groups"

Week 9: Hyperbolic groups have no Z^2 subgroups; Gromov boundaries

Suggested reading: Löh 8.3

Suggested exercises: Löh 5.E.23, 8.E.14, 8.E.17 (fill in details left out in class), 8.E.25, 8.E.27, then related exercises 7.E.8,9,10 and 8.E.21

Week 8: Hyperbolic groups, basic properties

Suggested reading: Löh 7.3-7.5

Suggested exercises: Löh 7.E.8, 7.E.9, 7.E.10, 7.E.19, 7.E.23, 7.E.28

Week 7: Hyperbolic plane, hyperbolic metric spaces

Suggested reading: Löh Appendix A.3, 7.2 (and 7.1 for background)

Suggested exercises: Verify computations left as exercises in class Monday and Wednesday.  Löh 7.E.4, 7.E.5, 7.E.6

Week 6: Group actions, ping pong, Tits alternative

Suggested reading: Löh 4.2, 4.3, 4.4

Suggested exercises: 4.E.7, 4.E.16, 4.E.20

Week 5: Growth series and subgroup growth, amalgamated products and HNN extensions

Suggested reading: Löh 2.3

Suggested exercises: 2.E.32, 2.E.38

Week 4: Nilpotent and solvable groups, Gromov's theorem on polynomial growth

Suggested reading: Löh 6.1, 6.2,  6.3

Suggested exercises: 6.E.6, 6.E.10,  6.E.16, 6.E.17, 6.E.18

Week 3: Proof of Švarc–Milnor, applications, group growth

Suggested reading: Löh 5.4, 5.6

Suggested exercises: 5.E.16, 23 (and for more fun, 5.E.15 and 5.E.30)

Week 2: Cayley graphs examples, quasi-isometry, Švarc–Milnor lemma

Suggested reading: Löh 3.1, 3.2, 5.1, 5.2, 5.3 (and 4.1 if necessary). 

Suggested exercises 3.E.15, 19;   5.E.3, 4, 5

Week 1: Generating groups, group presentations, Cayley graphs

Suggested reading: Löh 2.1 and 2.2

Suggested exercises: 2.E.12, 14, 18, 19, 21, 27