Quasi-isometrically rigid subgroups in right-angled Coxeter groups

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 ${\Lambda }_1\le {\Gamma }_1$ there exists a minsquare subgraph ${\Lambda }_2\le {\Gamma }_2$ 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.

35 pages, 8 figures. Comments are welcome!

References in corpus (1)

• On the geometry of van Kampen diagrams of graph products of groups

Cited by in corpus (2)

• Quasi-isometric rigidity of subgroups and Filtered ends
• Quasi-isometry invariance of relative filling functions