Property FW and wreath products of groups: A simple approach using Schreier graphs

Abstract

The group property FW stands in-between the celebrated Kazhdan’s property (T) and Serre’s property FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended.

It follows from the work of Y. Cornulier that a finitely generated wreath product $G{\wr }_{X}H$ has property FW if and only if both $G$ and $H$ have property FW and $X$ is finite. The aim of this paper is to give an elementary, direct and explicit proof of this fact using Schreier graphs.