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Quasi-isometrically rigid subgroups in right-angled Coxeter groups

Anthony Genevois

Algebraic & Geometric Topology 22 (2022) 657–708

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(Γ1) and C(Γ2) are quasi-isometric, then for any minsquare subgraph Λ1 Γ1, there exists a minsquare subgraph Λ2 Γ2 such that the right-angled Coxeter groups C(Λ1) and C(Λ2) 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.

graph products, right-angled Coxeter groups, quasi-median graphs, large-scale geometry, Morse subgroups
Mathematical Subject Classification 2010
Primary: 20F65, 20F67, 20F69
Received: 26 October 2019
Revised: 4 June 2020
Accepted: 11 January 2021
Published: 3 August 2022
Anthony Genevois
Institut Montpelliérain Alexander Grothendieck
Université de Montpellier