A new proof of Gromov's theorem on groups of polynomial growth

Bruce Kleiner, Yale University

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.

I will explain the proof in a manner accessible to a broad audience of
topologists, geometers, and analysts.