On the coarse geometry of certain right-angled Coxeter groups

Abstract

Let $\mathrm{\Gamma }$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\mathrm{\Gamma }$ is $\mathcal{C}\mathcal{ℱ}\mathcal{S}$, we prove that the right-angled Coxeter group $G_{\mathrm{\Gamma }}$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these groups. Furthermore, we prove that $G_{\mathrm{\Gamma }}$ is hyperbolic relative to a collection of $\mathcal{C}\mathcal{ℱ}\mathcal{S}$ right-angled Coxeter subgroups of $G_{\mathrm{\Gamma }}$. Consequently, the divergence of $G_{\mathrm{\Gamma }}$ is linear, 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.