601B -- Introduction to Geometric Group Theory (Fall 2023)
Course Info
Who: Prof Daniel Studenmund
What: Math 601B -- Introduction to Geometric Group Theory
When: 2:20-3:20 MWF
Where: Whitney 100E
Why: To provide a reasonably accessible introduction to an area of math with many connections to geometry, topology, and algebra
Office: Whitney 114
Office Hours: Mondays and Wednesdays after class, and any time by appointment
Primary Book: Geometric Group Theory: an Introduction, by Löh (draft pdf, errata)
Recommended resources: Geometric Group Theory, by Druţu and Kapovich
Office hours with a geometric group theorist, eds. Clay and Margalit (errata)
Metric spaces of nonpositive curvature, by Bridson and Haefliger
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