A new upper bound for the asymptotic dimension of RACGs (Q6634706)

Let \(W_{\Gamma}\) be the right-angled Coxeter group with defining graph \(\Gamma=(V(\Gamma), E(\Gamma))\). An isometric embedding theorem by \textit{T. Januszkiewicz} [Fundam. Math. 174, No. 1, 79--86 (2002; Zbl 1038.20025)] shows that Coxeter groups have finite asymptotic dimension \(\textrm{asdim}\) and \(\textrm{asdim}(W_{\Gamma}) \leq \# V(\Gamma)\).\N\NThe clique-connected dimension of a finite graph, \(\mathrm{dim}_{CC}(\Gamma)\), can be described as an index an index showing how connected is the graph modulo cliques. For example, if \(\Gamma\) is a clique, then \(\mathrm{dim}_{CC}(\Gamma)=0\).\N\NThe main results in the paper under review are the following.\N\NTheorem 1.1: Let \(W_{\Gamma}\) be the RACG with connected defining graph \(\Gamma\). Then \(\textrm{asdim}(W_{\Gamma}) \leq \mathrm{dim}_{CC}(\Gamma)\). If \(\Gamma\) is not connected, then \(\textrm{asdim}(W_{\Gamma}) \leq \max \{1, \mathrm{dim}_{CC}(\Gamma) \}\).\N\NProposition 6.2: Let \(W_{\Gamma}\) be the RACG with connected defining graph \(\Gamma\). If \(\mathrm{dim}_{CC}(\Gamma) \leq 2\), then \(\textrm{asdim}(W_{\Gamma})= \mathrm{dim}_{CC}(\Gamma)\).\N\NTheorem 1.2: Let \(W_{\Gamma}\) be the RACG with connected defining graph \(\Gamma\). Then \(\textrm{asdim}(W_{\Gamma}) \leq \min \{\mathrm{dim}_{CC}(\Gamma), \mathrm{dim} \Sigma(W_{\Gamma}) \}\). If \(\Gamma\) is not connected, then \(\mathrm{asdim} W_{\Gamma} \leq \max \{1, \min \{ \mathrm{dim}_{CC}(\Gamma), \mathrm{dim} \Sigma(W_{\Gamma}) \} \}\).\N\NThe author also generalizes some of these results to graph products of finite groups.