Group properties characterised by configurations

Abstract

J. M. Rosenblatt and G. A. Willis introduced the notion of configurations for finitely generated groups $G$. They characterised amenability of $G$ in terms of the configuration equations. In this paper we investigate which group properties can be characterised by configurations. It is proved that if $G_1$ and $G_2$ are two finitely generated groups having the same configuration sets and $G_1$ satisfies a semigroup law, then $G_2$ satisfies the same semigroup law. Furthermore, if $G_1$ is abelian then $G_1$ and $G_2$ are isomorphic.