On the coarse geometry of certain right-angled Coxeter groups

Nguyen, Hoang; Tran, Hung

doi:10.2140/agt.2019.19.3075

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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

DOI: 10.2140/agt.2019.19.3075

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 $C ℱ S$ , we prove that the right-angled Coxeter group $G_{Γ}$ 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 $G_{Γ}$ is hyperbolic relative to a collection of $C ℱ S$ right-angled Coxeter subgroups of $G_{Γ}$ . Consequently, the divergence of $G_{Γ}$ 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.

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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

Volume 19, issue 6 (2019)