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

Cora Welsch

Published 2016-03-22Version 1

In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph $\Gamma(H)$. For a finitely generated group $G$ we consider the poset $P_{\text{fi}}(G)$ of all right cosets of all proper finite index subgroups of $G$. We use the theory of subgroup graphs to prove that for many finitely generated infinite groups the order complex $\Delta P_{\text{fi}} (G)$ and the corresponding nerve complex are contractible.