Mathematics

Hyperbolicity in geometric group theory

Heejoung Kim (Illinois Math)

Abstract: In geometric group theory, we study a finitely generated group by looking at how the group acts on a metric space, using the topological or geometric properties of the metric space to shed light on the group. In particular, there has been lots of study about a finitely generated group acting nicely on a hyperbolic space based on properties of hyperbolic geometry. However, an arbitrary group does not admit a nice action on a hyperbolic space in general. Hence, many people have tried to generalize the techniques of hyperbolic geometry to study a more general metric space where a group might act nicely. In this talk, we will discuss hyperbolic geometry and how to use it to understand a finitely generated group. Moreover, we will talk about some generalizations of hyperbolic geometry with examples.