A new upper bound for the Asymptotic Dimension of RACGs

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