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

TL;DR

This paper extends subgroup graph theory to finite index subgroups of finitely generated groups, demonstrating that certain associated complexes are contractible, which advances understanding of their topological properties.

Contribution

It generalizes subgroup graph theory to finite index subgroups of finitely generated groups and analyzes the topological properties of related complexes.

Findings

Order complex and nerve complex are contractible for many finitely generated infinite groups.

Properties of subgroup graphs relate to the structure of finite index subgroups.

Provides new tools for studying the topology of subgroup posets.

Abstract

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.