Asymptotic dimension of Artin groups and a new upper bound for Coxeter\n groups

2022/03/10 by Panagiotis Tselekidis, Tselekidis, Panagiotis
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2203.05746

openalex publication_date 2022/03/10 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

If A_\Γ (W_\Γ) is the Artin (Coxeter) group with defining graph\n\Γ we denote by Sim(\Γ) the number of vertices of the largest\nclique in \Γ. We show that asdimA_\Γ \≤ Sim(\Γ), if\nSim(\Γ)=2. We conjecture that the inequality holds for every Artin group.\nWe prove that if for all free of infinity Artin (Coxeter) groups the conjecture\nholds, then it holds for all Artin (Coxeter) groups. As a corollary, we show\nthat asdimW_\Γ \≤ Sim(\Γ) for all Coxeter groups, which is the best\nknown upper bound for the asymptotic dimension of Coxeter Groups. As a further\ncorollary, we show that the asymptotic dimension of any Artin group of large\ntype with Sim(\Γ)=3 is exactly two.\n

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