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

Abstract: The group property FW stands in-between the celebrated Kazdhan'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 $G\wr_XH$ 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.