Let be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph is , we prove that the right-angled Coxeter group 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 is hyperbolic relative to a collection of right-angled Coxeter subgroups of . Consequently, the divergence of 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.
"On the coarse geometry of certain right-angled Coxeter groups." Algebr. Geom. Topol. 19 (6) 3075 - 3118, 2019. https://doi.org/10.2140/agt.2019.19.3075