Reflection trees of graphs as boundaries of Coxeter groups

Świątkowski, Jacek

doi:10.2140/agt.2021.21.351

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Reflection trees of graphs as boundaries of Coxeter groups

Jacek Świątkowski

Algebraic & Geometric Topology 21 (2021) 351–420

DOI: 10.2140/agt.2021.21.351

Abstract

To any finite graph $X$ (viewed as a topological space) we associate an explicit compact metric space $𝒳^{r} (X)$ , which we call the reflection tree of graphs $X$ . This space is of topological dimension $\leq 1$ 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 $\partial_{\infty} (W, S)$ of the right-angled Coxeter system $(W, S)$ with the nerve isomorphic to $Γ$ is homeomorphic to $𝒳^{r} (X)$ . For each $X$ , this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space $𝒳^{r} (X)$ .

I dedicate this paper to the memory of my parents.

Keywords

Coxeter group, visual boundary, hyperbolic group, Gromov boundary

Mathematical Subject Classification 2010

Primary: 20F65, 20F67

Secondary: 20F55, 57M07

References

Publication

Received: 27 August 2019

Accepted: 15 June 2020

Published: 25 February 2021

Authors

Jacek Świątkowski
Instytut Matematyczny Uniwersytet Wrocławski Wrocław Poland

Volume 21, issue 1 (2021)