Ergodic theorems for affine actions of amenable groups on Hilbert space

TL;DR

This paper establishes new ergodic theorems for affine actions of amenable groups on Hilbert spaces, leading to a proof of Gromov's theorem using ergodic theory techniques.

Contribution

It proves a weak mean ergodic theorem for 1-cocycles of amenable groups and applies it to affine actions, providing a new proof of Gromov's theorem.

Findings

Weak mean ergodic theorem for 1-cocycles of amenable groups

Almost fixed points for affine actions with weakly mixing linear parts

Elementary ergodic proof of Gromov's theorem

Abstract

We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. Let $G$ be a finitely generated, discrete, amenable group $G$ which admits a controlled Folner sequence. We use Theorem A to deduce that any affine action $G\ca^T \Cal H$ on Hilbert space with weakly mixing linear part admits a sequence of almost fixed points (Theorem B). Specializing to the case that $G$ is a finitely generated group of polynomial growth, we show that convex combinations of averages of the associated 1-cocycle over $n$-balls provide a sequence of almost fixed points for the action $G\ca^T \Cal H$ (Corollary C). This affirms a weak form of a conjecture of Shalom independently of Gromov's theorem on the virtual nilpotency of groups of polynomial growth. As a consequence, we are able to give a new, elementary, ergodic-theoretical proof of Gromov's theorem.