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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 CℱS, 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 CℱS 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
Nguyen, Hoang 1 ; Tran, Hung 2
@article{10_2140_agt_2019_19_3075,
author = {Nguyen, Hoang and Tran, Hung},
title = {On the coarse geometry of certain right-angled {Coxeter} groups},
journal = {Algebraic and Geometric Topology},
pages = {3075--3118},
publisher = {mathdoc},
volume = {19},
number = {6},
year = {2019},
doi = {10.2140/agt.2019.19.3075},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.3075/}
}
TY - JOUR AU - Nguyen, Hoang AU - Tran, Hung TI - On the coarse geometry of certain right-angled Coxeter groups JO - Algebraic and Geometric Topology PY - 2019 SP - 3075 EP - 3118 VL - 19 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.3075/ DO - 10.2140/agt.2019.19.3075 ID - 10_2140_agt_2019_19_3075 ER -
%0 Journal Article %A Nguyen, Hoang %A Tran, Hung %T On the coarse geometry of certain right-angled Coxeter groups %J Algebraic and Geometric Topology %D 2019 %P 3075-3118 %V 19 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.3075/ %R 10.2140/agt.2019.19.3075 %F 10_2140_agt_2019_19_3075
Nguyen, Hoang; Tran, Hung. On the coarse geometry of certain right-angled Coxeter groups. Algebraic and Geometric Topology, Tome 19 (2019) no. 6, pp. 3075-3118. doi: 10.2140/agt.2019.19.3075
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