2017/12/31 by Hoang Thanh Nguyen, Hung Cong Tran, Hung V. Tran · 10 citations
Mathematics · #Advanced Combinatorial Mathematics #Artin group #Combinatorics #Coxeter complex #Coxeter graph #Coxeter group #Geometric and Algebraic Topology #Geometry #Graph #Homotopy and Cohomology in Algebraic Topology #Line graph #Longest element of a Coxeter group #Mathematics #Point group #Regular polygon #Uniform k 21 polytope #Voltage graph #math.GR #math.GT
paper · pdf · doi:10.2140/agt.2019.19.3075
published in Algebraic & Geometric Topology 19(6), 3075-3118 (Mathematical Sciences Publishers) · 38 pages, 6 figures. Minor changes and other updates to incorporate referee comments. To appear in Algebraic & Geometric Topology. arXiv admin note: text overlap with arXiv:1708.07818
arxiv created 2019/03/04 · openalex publication_date 2019/10/20 · arxiv updated 2019/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let [math] be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph [math] is [math] , we prove that the right-angled Coxeter group [math] 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 [math] is hyperbolic relative to a collection of [math] right-angled Coxeter subgroups of [math] . Consequently, the divergence of [math] 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.