Let be a finitely presented group, and its commutator subgroup. Let be the Cayley graph of with all commutators in as generators. Then is large scale simply connected. Furthermore, if is a torsion-free nonelementary word-hyperbolic group, is one-ended. Hence (in this case), the asymptotic dimension of is at least .
"Large scale geometry of commutator subgroups." Algebr. Geom. Topol. 8 (4) 2131 - 2146, 2008. https://doi.org/10.2140/agt.2008.8.2131