Math 194 Spring 2021

An Introduction to the Geometry of Groups

Textbook for the class

Meeting Times

Once per week on zoom. See iLearn for zoom link.

Time: Fridays, 2pm

Office Hours: Tuesdays, 1pm

Rough Course Outline

This is a reading course, and not a lecture course. Each week, you’ll read parts of the textbook corresponding to what’s assigned below, and then during our meeting we’ll discuss whatever parts of the section you found most interesting / most difficult. Doing exercises from the text are encouraged for your understanding, but will not be required.


  1. Chapter 1, review of groups and generating sets

  2. Chapter 2.1, Cayley Graphs

  3. Chapter 2.2, Metric Graphs and Metric Spaces

  4. Exercise session

  5. Chapter 3, Group actions on Trees and Free actions

  6. Chapter 3, Group actions on Trees and Free actions

  7. Chapter 7, Quasi-isometries

  8. Chapter 7, Quasi-isometries and an intro to the Milnor-Schwartz Lemma

  9. Chapter 7, proof of the Milnor-Schwartz Lemma

  10. Exercise session and final presentation practice

  11. Final presentation (Topic Goal: Explain the assumptions of the Milnor-Schwartz Lemma, and outline a proof)


About the Course

When is there a relationship between the group structure of a group G, and the metric structure of a metric space X? Can we turn a group into a metric space? Can we turn a metric space into a group? If we can, what are the relationships between the two? These questions lie at the heart of geometric group theory, whose goal is to study groups via the metric spaces they act (nicely) on, and to study metric spaces via the group of isometries on the space.


The beginning of the course will be used to build up some familiarity with the above ideas, and the focus will be on constructing geometric spaces which correspond to groups, and understanding how the geometric space encodes the original group structure. The second half of the course will focus on a natural equivalence class for these geometric structures called quasi-isometry. Finally, the course will culminate in a discussion of the Milnor-Schwartz Lemma (aka, "the Fundamental Theorem of Geometric Group Theory"), which shows that the coarse geometry of a finitely generated group is independent of the choice of generating set, i.e., we can discuss the geometry of a group as opposed to a geometry for a group.

Additional Resources