#### Volume 22, issue 2 (2022)

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Quasi-isometrically rigid subgroups in right-angled Coxeter groups

### Anthony Genevois

Algebraic & Geometric Topology 22 (2022) 657–708
##### 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 $C\left({\Gamma }_{\phantom{\rule{-0.17em}{0ex}}1}\right)$ and $C\left({\Gamma }_{\phantom{\rule{-0.17em}{0ex}}2}\right)$ are quasi-isometric, then for any minsquare subgraph ${\Lambda }_{1}\le {\Gamma }_{\phantom{\rule{-0.17em}{0ex}}1}$, there exists a minsquare subgraph ${\Lambda }_{2}\le {\Gamma }_{\phantom{\rule{-0.17em}{0ex}}2}$ such that the right-angled Coxeter groups $C\left({\Lambda }_{1}\right)$ and $C\left({\Lambda }_{2}\right)$ 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