Large scale geometry of commutator subgroups

Abstract

Let $G$ be a finitely presented group, and $G^{\prime }$ its commutator subgroup. Let $C$ be the Cayley graph of $G^{\prime }$ with all commutators in $G$ as generators. Then $C$ is large scale simply connected. Furthermore, if $G$ is a torsion-free nonelementary word-hyperbolic group, $C$ is one-ended. Hence (in this case), the asymptotic dimension of $C$ is at least $2$.