PSU Mathematics Department - Meeting Details

For more information about this meeting, contact Svetlana Katok, Anatole Katok.

Abstract:
Title:	A new proof of Gromov's theorem on groups of polynomial growth
Seminar:	Center for Dynamics and Geometry Seminar
Speaker:	Bruce Kleiner, Yale
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. The main step involves showing that any infinite group of polynomial growth admits a finite dimensional linear representation with infinite image. We establish this using harmonic maps, thereby avoiding the Montgomery-Zippin-Yamabe theory of locally compact groups which was used in Gromov's original proof.