Math 4803

Intro. Geometric Group Theory

Spring 2021
-

Syllabus

 Click here.

Class Meetings

 Tuesday, Thursday, 3:30-4:45 pm on Microsoft Teams.

Textbook

Office Hours

 Tue 11-12, Fri 2-3 and by appointment, on Teams.

Resources

 Final project information
Peer paper evaluation
Final paper evaluation

Schedule
Week Dates Topics Text Sections Homework Notes Comments
1 Jan 14 Intro Jan 14 First day of class
2 Jan 19/21 Groups OHGGT 1 A.1, A.2*, A.3* Jan 19
Jan 21		 Deadline for schedule changes
3 Jan 26/28 Cayley's theorems 1.1-1.5 Chap 1: 6, 7, 12, 20, 22 Jan 26
Jan 28
4 Feb 2/4 Cayley graphs 1.6-1.8 Chap 1: 14, 15, 18, 21, A.4* Feb 2
Feb 4		 Topic/group due
5 Feb 9/11 Coxeter groups 2 A.5, Chap 2: 3(a-c), 3(d-f)*, 6, 9 Feb 9
Feb 11
6 Feb 16/18 Free groups 3.1-3.3 Chap 3: 8, 9, 10, 11 Feb 16
Feb 18
7 Feb 23/25 Free products 3.4-3.6 Chap 3: 20(a), 24, A.6 Feb 23
Feb 25		 Abstract due
8 Mar 2/4 Baumslag-Solitar groups 4 Midterm Mar 2
Mar 4		 Take-home midterm
9 Mar 9/11 The word problem 5 none Mar 9
Mar 11
10 Mar 18 Burnside's problem 6 Chap 5: 9, Chap 6: 4, 5 Mar 11 Spring break
Grade mode / withdraw deadline
11 Mar 23/25 Automata 7 Chap 7: 1 Mar 23
Mar 25		 Spring break
12 Mar 30/32 Lamplighter & Thompson's groups 8, 10 A.7 or Chap 10: 4 Mar 30
Apr 1
Outline due
13 Apr 6/8 Large scale properties 10 Chap 11: 13 Apr 6
Apr 8
Draft due
14 Apr 13/15 Quasi-isometries 11 A.8 Apr 13
15 Apr 20/22 Presentations none
16 Apr 27 Farewell Apr 27 Last day of class
Project due
16 GGT notes

Additional homework problems
  1. 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)
  2. (Optional) Use strand diagrams to verify the presentation of the symmetric group given in class.
  3. (Optional) Prove that free groups exist.
  4. (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.
  5. 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.
  6. Prove that PSL_2(Z) is isomorphic to the free product of Z/2 with Z/3.
  7. Prove that the Cayley graph of the lamplighter group has a dead end. Show that it has dead ends of arbitrary depth.
  8. (Optional) Prove that quasi-isometries have quasi-inverses, and hence that QI(X) is a group.

Reading prompts
  1. Jan 21: Cayley graph
  2. Jan 28: Fundamental domain
  3. Feb 4: Infinite dihedral group
  4. Feb 11: Ping pong lemma
  5. Feb 18: Free product
  6. Feb 25: Baumslag-Solitar group
  7. Mar 4: none
  8. Mar 11: none
  9. Mar 18: none
  10. Mar 25: Lamplighter group
  11. Apr 1: Freudenthal-Hopf theorem
  12. Apr 8: Ends of groups

References

 ---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