Quasi-isometrically rigid subgroups in right-angled Coxeter groups

TL;DR

This paper identifies a class of quasi-isometrically rigid subgroups in right-angled Coxeter groups and explores their implications for quasi-isometry classifications, providing new insights into subgroup structures and hyperbolicity.

Contribution

It introduces eccentric subgroups as quasi-isometrically rigid in graph products and establishes a correspondence between quasi-isometric right-angled Coxeter groups and their subgraphs.

Findings

Eccentric subgroups are quasi-isometrically rigid.

Quasi-isometry of Coxeter groups implies quasi-isometry of certain subgroups.

Characterization of when Morse subgroups are hyperbolic.

Abstract

In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple class of quasi-isometrically rigid subgroups in graph products of finite groups, which we call eccentric subgroups. As an application, we prove that, if two right-angled Coxeter groups $C(\Gamma_1)$ and $C(\Gamma_2)$ are quasi-isometric, then for any minsquare subgraph $\Lambda_1 \leq \Gamma_1$ there exists a minsquare subgraph $\Lambda_2 \leq \Gamma_2$ such that the right-angled Coxeter groups $C(\Lambda_1)$ and $C(\Lambda_2)$ are quasi-isometric as well. Various examples of non-quasi-isometric groups are deduced. Our arguments are based on a study of non-hyperbolic Morse subgroups in graph products of finite groups. As a by-product, we are able to determine precisely when a right-angled Coxeter group has all its infinite-index Morse subgroups hyperbolic, answering a question of Russell, Spriano and Tran.