Proper actions of lamplighter groups associated with free groups

TL;DR

This paper proves that wreath products of finite groups with free groups can act properly and isometrically on Hilbert spaces, advancing understanding of their geometric properties.

Contribution

It establishes that lamplighter groups built from free groups admit proper isometric actions on Hilbert spaces, a novel result in geometric group theory.

Findings

Wreath products of finite groups with free groups admit proper isometric actions on Hilbert spaces.

This extends the class of groups known to have such actions, impacting the study of a-T-menability.

The result has implications for the Baum-Connes conjecture and related areas.

Abstract

Given a finite group $H$ and a free group $F_n$, we prove that the wreath product $H\wr F_n$ admits a metrically proper, isometric action on a Hilbert space.