An Introduction to the Geometry of Groups
Office Hours with a Geometric Group Theorist, edited by Matt Clay and Dan Margalit
(Link to download through UCR Library)
Once per week on zoom. See iLearn for zoom link.
Time: Fridays, 2pm
Office Hours: Tuesdays, 1pm
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.
Chapter 1, review of groups and generating sets
Chapter 2.1, Cayley Graphs
Chapter 2.2, Metric Graphs and Metric Spaces
Exercise session
Chapter 3, Group actions on Trees and Free actions
Chapter 3, Group actions on Trees and Free actions
Chapter 7, Quasi-isometries
Chapter 7, Quasi-isometries and an intro to the Milnor-Schwartz Lemma
Chapter 7, proof of the Milnor-Schwartz Lemma
Exercise session and final presentation practice
Final presentation (Topic Goal: Explain the assumptions of the Milnor-Schwartz Lemma, and outline a proof)
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.
Another textbook: Geometric Group Theory: An Introduction by Clara Löh
Alessandro Sisto's Lecture notes on Geometric Group Theory