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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

Let Γ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph Γ is CS, we prove that the right-angled Coxeter group GΓ 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 GΓ is hyperbolic relative to a collection of CS right-angled Coxeter subgroups of GΓ. Consequently, the divergence of GΓ 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.

quasi-isometry, right-angled Coxeter group
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Received: 17 August 2018
Revised: 31 December 2018
Accepted: 3 March 2019
Published: 20 October 2019
Hoang Thanh Nguyen
Department of Mathematical Sciences
University of Wisconsin–Milwaukee
Milwaukee, WI
United States
Hung Cong Tran
Department of Mathematics
The University of Georgia
Athens, GA
United States