#### Volume 19, issue 6 (2019)

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On the coarse geometry of certain right-angled Coxeter groups

### Hoang Thanh Nguyen and Hung Cong Tran

Algebraic & Geometric Topology 19 (2019) 3075–3118
##### 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 $\mathsc{C}\mathsc{ℱ}\mathsc{S}$, 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 groups. Furthermore, we prove that ${G}_{\Gamma }$ is hyperbolic relative to a collection of $\mathsc{C}\mathsc{ℱ}\mathsc{S}$ right-angled Coxeter subgroups of ${G}_{\Gamma }$. Consequently, the divergence of ${G}_{\Gamma }$ 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.

##### Keywords
quasi-isometry, right-angled Coxeter group
##### Mathematical Subject Classification 2010
Primary: 20F65, 20F67