# Math 112: Geometric Group Theory

## Fall 2015

**General Information:**

Instructor: Bjoern Muetzel

E-Mail: bjorn.mutzel-at-dartmouth.edu

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.

__Contents:__

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:**

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.

**Literature:**

- 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