Quasi-isometrically rigid subgroups in right-angled Coxeter groups

Genevois, Anthony

doi:10.2140/agt.2022.22.657

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Quasi-isometrically rigid subgroups in right-angled Coxeter groups

Anthony Genevois

Algebraic & Geometric Topology 22 (2022) 657–708

DOI: 10.2140/agt.2022.22.657

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 $C (Γ_{1})$ and $C (Γ_{2})$ are quasi-isometric, then for any minsquare subgraph $Λ_{1} \leq Γ_{1}$ , there exists a minsquare subgraph $Λ_{2} \leq Γ_{2}$ such that the right-angled Coxeter groups $C (Λ_{1})$ and $C (Λ_{2})$ are quasi-isometric as well. Various examples of non-quasi-isometric groups are deduced. Our arguments are based on a study of nonhyperbolic 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.

Keywords

graph products, right-angled Coxeter groups, quasi-median graphs, large-scale geometry, Morse subgroups

Mathematical Subject Classification 2010

Primary: 20F65, 20F67, 20F69

References

Publication

Received: 26 October 2019

Revised: 4 June 2020

Accepted: 11 January 2021

Published: 3 August 2022

Authors

Anthony Genevois
Institut Montpelliérain Alexander Grothendieck Université de Montpellier Montpellier France

Volume 22, issue 2 (2022)