Reflection trees of graphs as boundaries of Coxeter groups

Abstract

To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\mathcal{X}}^r(X)$ which we call {\it 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 $\Gamma$ 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 $\Gamma$ is homeomorphic to ${\mathcal{X}}^r(X)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${\mathcal{X}}^r(X)$.

53 pages