A functional analysis proof of Gromov's polynomial growth theorem

TL;DR

This paper presents a new proof of Gromov's polynomial growth theorem for finitely generated groups, utilizing reduced cohomology and Shalom's property H_FD, offering an alternative to existing proofs.

Contribution

It introduces a novel proof approach based on cohomological analysis and property H_FD, expanding the methods available for understanding Gromov's theorem.

Findings

Provides a new proof of Gromov's theorem

Utilizes reduced cohomology and property H_FD

Enhances understanding of group growth and structure

Abstract

The celebrated theorem of Gromov asserts that any finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner and others. In this note, we give yet another proof of Gromov's theorem, along the lines of Shalom and Chifan--Sinclair, which is based on the analysis of reduced cohomology and Shalom's property H_FD.