A new upper bound for the Asymptotic Dimension of RACGs

Abstract

Let $W_{\Gamma }$ be the Right-Angled Coxeter group with defining graph $\Gamma$. We show that the asymptotic dimension of $W_{\Gamma }$ is smaller than or equal to $di{m}_{CC}(\Gamma )$, the clique-connected dimension of the graph. As a corollary we show that $W_{\Gamma }$ is virtually free if and only if $di{m}_{CC}(\Gamma )=1$.

14 pages