Math 112: Geometric Group Theory

Fall 2015

General Information:

Instructor: Bjoern Muetzel
Office: 317 Kemeny Hall
Lectures: MWF 12:30–1:35
Problem sessions: Tu 1:00–1:50
Room: 004 Kemeny Hall
Office hours: Tu, Th 10–11:30 or by appointment

Course description:
By associating a group to one of its Cayley graphs, the properties of a group can be studied from a geometric point of view. The group itself acts on the Cayley graph via isometries which is reflected in the symmetries of the graph. The inherent beauty of a group can thus be visualized in its Cayley graph making these graphs a fascinating object of study.
Geometric group theory examines the connection between geometric and algebraic invariants of a group. In order to obtain interesting invariants one usually restricts oneself to finitely generated groups and takes invariants from large scale geometry.
Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry and differential geometry and has numerous applications to problems in classical fields, like combinatorial
group theory, graph theory and differential topology.

Topics: Graphs and trees, Cayley graphs, free groups, hyperbolic groups, large scale geometry.

Chapter I - Graphs and trees
Chapter II - Cayley graphs
Chapter III - Geometric realizations of graphs
Chapter IV - Finitely generated groups
Chapter V - Geometry from far away
Chapter VI - Space of ends
Chapter VII - Hyperbolic groups


Homework problems will be assigned weekly and discussed in the next problem session. Collaboration on homework is permitted and encouraged. Any resource is allowed provided you reference it. But you must write up your solutions by yourself.

Questionnaire 1
Homework 1
Homework 2
Homework 3 for exercise 6 see: A Farey Tale
Homework 4
Homework 5
Homework 6 solution for exercise 16: exercise 16
Homework 7 solution for exercise 21: exercise 21
Homework 8 solution for exercise 22,23: exercise 22,23
Homework 9

Special considerations:

Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see their instructor as soon as possible.


- Brian Bowditch: A course on geometric group theory, lecture notes
- Cornelia Drutu, Michael Kapovich: Lectures on geometric group theory
- Clara Loeh: Geometric group theory ‐ an introduction, lecture notes
(these three can be found via Google and downloaded for free)
- Ghys, Haefliger and Verjovsky: Group theory from a geometrical viewpoint

Background reading:

- Bridson, Haefliger: Metric spaces of non-positive curvature
- Diestel: Graph theory
- Hatcher: Algebraic topology
- Lyndon, Schupp: Combinatorial group theory
- Serre: Trees