Reflection trees of graphs as boundaries of Coxeter groups

Abstract

To any finite graph $X$ (viewed as a topological space) we associate an explicit compact metric space ${\mathcal{𝒳}}^r\left(X\right)$, which we call the reflection tree of graphs $X$. 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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ is homeomorphic to ${\mathcal{𝒳}}^r\left(X\right)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${\mathcal{𝒳}}^r\left(X\right)$.