On the coarse geometry of certain right-angled Coxeter groups

TL;DR

This paper classifies the large-scale geometric structure of certain right-angled Coxeter groups based on properties of their defining graphs, revealing their quasi-isometry types and divergence behaviors.

Contribution

It provides a complete quasi-isometry classification for these groups and extends the analysis to high-dimensional cases, introducing new geometric insights.

Findings

Groups are virtually Seifert or graph manifold groups if the graph is CFS.

Otherwise, groups are hyperbolic relative to certain subgroups with linear, quadratic, or exponential divergence.

High-dimensional analogs of virtually graph manifold groups are constructed and studied.

Abstract

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_\Gamma$. Consequently, the divergence of $G_\Gamma$ is linear, or quadratic, or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex torsion free infinite index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.