# Geometric Analysis and Topology Seminar

#### A New Proof of Gromov's Theorem on Groups of Polynomial Growth

**Speaker:**
Bruce Kleiner, Yale

**Location:**
Warren Weaver Hall 613

**Date:**
Friday, November 30, 2007, 1 p.m.

**Synopsis:**

In 1981 Gromov showed that any finitely generated group of polynomial growth contains a finite index nilpotent subgroup. This was a landmark paper in several respects. The proof was based on the idea that one can take a sequence of rescalings of an infinite group G, pass to a limiting metric space, and apply deep results about the structure of locally compact groups to draw conclusions about the original group G. In the process, the paper introduced Gromov-Hausdorff convergence, initiated the subject of geometric group theory, and gave the first application of the Montgomery-Zippin solution to Hilbert's fifth problem (and subsequent extensions due to Yamabe).

The purpose of the lecture is to give a new, much shorter, proof of Gromov's theorem which is based on analysis instead of the Montgomery-Zippin-Yamabe theory. We will try to make the lecture as broadly accesssible as possible.