# Reflection trees of graphs as boundaries of Coxeter groups

## TL;DR

This paper introduces the reflection tree space associated with finite graphs and demonstrates its homeomorphism to the visual boundary of certain right-angled Coxeter groups, linking graph topology to group boundaries.

## Contribution

It constructs explicit boundary spaces called reflection trees for graphs and establishes their homeomorphism to Coxeter group boundaries, providing new insights into geometric group theory.

## Key findings

- Reflection trees are compact, topologically dimension ≤1 spaces.
- Homeomorphism between reflection trees and Coxeter group boundaries.
- Many hyperbolic groups have boundaries homeomorphic to reflection trees.

## Abstract

To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\cal X}^r(X)$ which we call {\it the reflection tree of graphs $X$}. This space is of topological dimension $\le1$ 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 ${\cal X}^r(X)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${\cal X}^r(X)$.