Algebraic & Geometric Topology

On the coarse geometry of certain right-angled Coxeter groups

Hoang Thanh Nguyen and Hung Cong Tran

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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.

Article information

Algebr. Geom. Topol., Volume 19, Number 6 (2019), 3075-3118.

Received: 17 August 2018
Revised: 31 December 2018
Accepted: 3 March 2019
First available in Project Euclid: 29 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups

quasi-isometry right-angled Coxeter group


Nguyen, Hoang Thanh; Tran, Hung Cong. On the coarse geometry of certain right-angled Coxeter groups. Algebr. Geom. Topol. 19 (2019), no. 6, 3075--3118. doi:10.2140/agt.2019.19.3075.

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