Groups and Geometry
The aim of the course is to introduce students to the principle that groups can be understood through the geometry of spaces associated with them. By studying Cayley graphs, trees, and the hyperbolic plane, students will develop geometric intuition for abstract algebraic structures and gain an accessible introduction to ideas from geometric group theory, topology, and hyperbolic geometry.
By the end of the course, a successful student should be able to:
1. Construct Cayley graphs of basic finitely generated groups and use the word metric.
2. Understand trees and describe simple group actions on them.
3. Explain the basic ideas of free products, amalgamated products, HNN extensions, and Bass–Serre trees.
4. Work with the upper half-plane models of the hyperbolic plane, including geodesics and basic isometries.
5. Understand isometric group actions on the hyperbolic plane.
These outcomes will be assessed through weekly homework, a project, and two exams.
This course introduces geometric methods for studying groups, with an emphasis on Cayley graphs, trees, Bass–Serre theory, and the hyperbolic plane. The central idea is that algebraic properties of a group can often be understood by studying a geometric space associated with the group or a space on which the group acts.
The course begins with finitely generated groups, generators and relations, Cayley graphs, and word metrics. Concrete examples will include cyclic groups, free abelian groups, free groups, and groups defined by simple presentations.
We also study trees and group actions on trees. This leads to an introduction to Bass–Serre theory, including free products, amalgamated free products, and HNN extensions.
The course also covers the geometry of the hyperbolic plane. Topics include the upper half-plane models, geodesics, hyperbolic distance, isometries. We will also discuss discrete groups acting on the hyperbolic plane.
1. Basic group theory: groups, generators, relations, and examples. Cayley graphs and word metrics.
2. Trees, Group actions on trees. Free products, amalgamated products, and HNN extensions. Basic ideas of Bass–Serre theory.
3. Models of the hyperbolic plane. Hyperbolic distance, geodesics, and hyperbolic triangles. Isometries. Discrete group actions and hyperbolic surfaces.
Weekly Homework: 20%
Project: 40%
Exam 1: 20%
Exam 2: 20%
Students should have completed standard undergraduate courses in linear algebra and single-variable calculus. Familiarity with basic proof techniques and elementary mathematical notation is expected.
Some previous exposure to abstract algebra is helpful but not required. The course will introduce the necessary definitions concerning groups and group actions. No prior knowledge of graph theory, Bass–Serre theory, algebraic topology, differential geometry, or hyperbolic geometry is assumed.
Students should be comfortable working with matrices and functions.
Jean-Pierre Serre, Trees, Springer.
Matt Clay and Dan Margalit, eds., Office Hours with a Geometric Group Theorist, Princeton University Press.
New for AY2026