Intro. Geometric Group Theory
Tuesday, Thursday, 3:30-4:45 pm on Microsoft Teams.
Tue 11-12, Fri 2-3 and by appointment, on Teams.
Final project information
Peer paper evaluation
Final paper evaluation
||First day of class
||A.1, A.2*, A.3*
|Deadline for schedule changes
||Chap 1: 6, 7, 12, 20, 22
||Chap 1: 14, 15, 18, 21, A.4*
||A.5, Chap 2: 3(a-c), 3(d-f)*, 6, 9
||Chap 3: 8, 9, 10, 11
||Chap 3: 20(a), 24, A.6
||The word problem
||Chap 5: 9, Chap 6: 4, 5
Grade mode / withdraw deadline
||Chap 7: 1
||Lamplighter & Thompson's groups
||A.7 or Chap 10: 4
||Large scale properties
||Chap 11: 13
||Last day of class
Additional homework problems
- Show that internal and external presentations of a group G are equivalent. More precisely, show that if we take an internal presentation for G and regard it as an external presentation of a group H (converting relations like ab=ba into relators like aba-1b-1 if needed), then H is isomorphic to G. And conversely, any external presentation of G naturally gives an internal presentation of G, once we identify the generators of the presentation with their images in G. As an example, take G to be Z2 with internal presentation < a,b : ab=ba >, where a=(1,0) and b=(0,1). The corresponding external presentation is < a,b : ab a-1b-1 >. In this case you would need to show that F_2 modulo the normal closure of a-1b-1 is isomorphic to Z2. For the other direction you need to show that < a,b : ab a-1b-1 > is an internal presentation for Z2. (Solution)
- (Optional) Use strand diagrams to verify the presentation of the symmetric group given in class.
- (Optional) Prove that free groups exist.
- (Optional) Suppose that G acts on a connected graph X with fundamental domain F and that H is a subgroup of G of index n. Show that there is a fundamental domain for H consisting of n copies of F.
- Prove that if G acts on a graph Γ with fundamental domain F, if the stabilizer of F in G is trivial, and if H has a fundamental domain consisting of n translates of F, then [G:H]=n. Hint: count the set of right cosets H\G. Optional: prove the converse: if [G:H]=n then there is a funamental domain for H consisting of n trainslates of F.
- Prove that PSL_2(Z) is isomorphic to the free product of Z/2 with Z/3.
- Prove that the Cayley graph of the lamplighter group has a dead end. Show that it has dead ends of arbitrary depth.
- (Optional) Prove that quasi-isometries have quasi-inverses, and hence that QI(X) is a group.
- Jan 21: Cayley graph
- Jan 28: Fundamental domain
- Feb 4: Infinite dihedral group
- Feb 11: Ping pong lemma
- Feb 18: Free product
- Feb 25: Baumslag-Solitar group
- Mar 4: none
- Mar 11: none
- Mar 18: none
- Mar 25: Lamplighter group
- Apr 1: Freudenthal-Hopf theorem
- Apr 8: Ends of groups
---Office hours with a geometric group theorist, Matt Clay and Dan Margalit
---Group theory videos: Actions Orbit-stabilizer theorem Group basics Normal subgroups Cayley's theorem