# Geometric Analysis and Topology Seminar

#### Asymptotic Cones of Groups

**Speaker:**
Mark Sapir, Vanderbilt

**Location:**
Warren Weaver Hall 517

**Date:**
Monday, February 23, 2015, 11 a.m.

**Synopsis:**

Asymptotic cones of groups are the major tools in studying "large scale" properties of finitely generated groups such as growth, isoperimetric functions, "representation varieties" of one group in another group, etc. In this talk I will survey the main applications of asymptotic cones. If the group grows polynomially (say, it is Abelian) asymptotic cones are Gromov-Hausdorff limits of rescaled Cayley graphs of the group (in a way, similar to obtaining Brownian motion from random walks). In general one needs ultraproducts to define asymptotic cones. In turn, ultraproducts connect geometry of groups with model theory which lead to very powerful result. For example, recent results of Hrushovsky imply that if one asymptotic cone of a group is locally compact, then the group is virtually nilpotent.