Geometric Group Theory: Negative Curvature in Group Theory

Math 835, Fall 2009

Geometric group theory is the study of finitely generated groups using geometry. Usually this means studying the geometry of the Cayley graph of a group with respect to some finite set of generators. This is a graph whose vertices are the group elements and whose edges correspond to multiplication on the right by a generator.

The local geometry of the Cayley graph is uninteresting and tells us little about the group. The large-scale geometry of the Cayley graph, on the other hand, can tell us a great deal about the group. In this class we'll focus on groups whose Cayley graph is Gromov hyperbolic; a space is Gromov hyperbolic if its large-scale geometry is "like" that of a tree. There are several ways to make this precise, and we'll spend a bit of time understanding these ways.

After the definitions and some intuition-building in trees and the hyperbolic plane, there are several possible directions we might take. I hope to give you lots of examples and to talk about at least some of the following topics:

• The boundary at infinity of a hyperbolic group.
• The asymptotic cone of a hyperbolic group.
• Algorithmic properties of hyperbolic groups.
• Finiteness properties of hyperbolic groups.
• Groups acting on CAT(0) and CAT(-1) spaces.
• Geometry of "random" groups.
• Relatively hyperbolic groups.
• Separability and residual finiteness.
• "Mesoscopic" curvature.

Prerequisites for the material we will cover are mild. You should know some basic group theory and point set topology.

References

• Bridson and Haefliger, Metric Spaces of Non-Positive Curvature, MR1744486, especially sections I.1, III.H, III.\Gamma.
• The "ABC..." notes, edited by Hamish Short.
• Coornaert, Delzant, and Papadopoulous, Géométrie et théorie des groupes, MR1075994.
• Ghys and de la Harpe editors, Sur les Groupes Hyperboliques d'après Mikhael Gromov.
• More to come...