Large scale geometry of commutator subgroups

TL;DR

This paper investigates the large-scale geometric properties of the commutator subgroup of finitely presented groups, revealing conditions under which the Cayley graph is simply connected, one-ended, and has high asymptotic dimension.

Contribution

It establishes that the Cayley graph of the commutator subgroup is large scale simply connected and, for certain hyperbolic groups, one-ended with high asymptotic dimension.

Findings

C is large scale simply connected

C is one-ended for torsion-free hyperbolic groups

Asymptotic dimension of C is at least 2

Abstract

Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' 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.