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Reflection trees of graphs as boundaries of Coxeter groups

Jacek Świątkowski

Algebraic & Geometric Topology 21 (2021) 351–420

To any finite graph X (viewed as a topological space) we associate an explicit compact metric space 𝒳r(X), which we call the reflection tree of graphs X. This space is of topological dimension 1 and its connected components are locally connected. We show that if X is appropriately triangulated (as a simplicial graph Γ for which X is the geometric realization) then the visual boundary (W,S) of the right-angled Coxeter system (W,S) with the nerve isomorphic to Γ is homeomorphic to 𝒳r(X). For each X, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space 𝒳r(X).

I dedicate this paper to the memory of my parents.

Coxeter group, visual boundary, hyperbolic group, Gromov boundary
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Secondary: 20F55, 57M07
Received: 27 August 2019
Accepted: 15 June 2020
Published: 25 February 2021
Jacek Świątkowski
Instytut Matematyczny
Uniwersytet Wrocławski