Volume 21, issue 1 (2021)

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Reflection trees of graphs as boundaries of Coxeter groups

Jacek Świątkowski

Algebraic & Geometric Topology 21 (2021) 351–420
Abstract

To any finite graph $X$ (viewed as a topological space) we associate an explicit compact metric space ${\mathsc{𝒳}}^{r}\left(X\right)$, which we call the reflection tree of graphs $X\phantom{\rule{-0.17em}{0ex}}$. This space is of topological dimension $\le 1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $\Gamma$ for which $X$ is the geometric realization) then the visual boundary ${\partial }_{\infty }\left(W,S\right)$ of the right-angled Coxeter system $\left(W,S\right)$ with the nerve isomorphic to $\Gamma$ is homeomorphic to ${\mathsc{𝒳}}^{r}\left(X\right)$. For each $X\phantom{\rule{-0.17em}{0ex}}$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${\mathsc{𝒳}}^{r}\left(X\right)$.

 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