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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 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.

Keywords
quasi-isometry, right-angled Coxeter group
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
References
Publication
Received: 17 August 2018
Revised: 31 December 2018
Accepted: 3 March 2019
Published: 20 October 2019
Authors
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