Subgroup graph methods for presentations of finitely generated groups and the contractibility of associated simplicial complexes

Abstract

In this article we generalize the theory of subgroup graphs of subgroups of free groups, developed by I. Kapovich and A. Myasnikov, based on a work by J. Stallings, to finite index subgroups of finitely generated groups. Given a presentation of a finitely generated group G=〈X|R〉 and a finite connected X-regular graph Γ which fulfills the relators R, we associate to Γ a finite index subgroup H of G. Conversely, the Schreier coset graph of H with respect to X and G is such a graph Γ. Firstly, we study and prove various properties of H in relation to the graph Γ. Secondly, we prove that for many finitely generated infinite groups the order and nerve complexes that we associate to G are contractible. In particular, this is the case for free and free abelian groups, Fuchsian groups of genus g≥2, infinite right angled Coxeter groups, Artin and pure braid groups, infinite virtually cyclic groups, Baumslag–Solitar groups as well as the (free) product of at least two of these, and all finite index subgroups of these groups.