Bowditch taut spectrum and dimensions of groups (Q6590155)

Let \(G\) be a discrete group and \(\mathrm{cd}_{R}(G)\) denote the cohomological dimension of \(G\) with respect to the ring \(R\). Similarly, let \(\underline{\mathrm{cd}}(G)\) and \(\underline{\underline{\mathrm{cd}}}(G)\) denote the proper cohomological dimension and the virtually cyclic cohomological dimension of \(G\), respectively, and let \(\underline{\mathrm{gd}}(G)\) and \(\underline{\underline{\mathrm{gd}}}(G)\) be their geometric counterparts.\N\NThe main result of this paper is Theorem A: Let \(\mathcal{G}\) denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups: \N\begin{enumerate}\N\item \(\big \{ G \in \mathcal{G} \big | \underline{\mathrm{cd}}(G)=2 \mbox{ and } \underline{\mathrm{gd}}(G)=3 \big \}\); \N\item \(\big \{ G \in \mathcal{G} \big | \underline{\underline{\mathrm{cd}}}(G)=2 \mbox{ and } \underline{\underline{\mathrm{gd}}}(G)=3 \big \}\); \N\item \(\big \{ G \in \mathcal{G} \big | \mathrm{cd}_{\mathbb{Q}}(G)=2 \mbox{ and } \mathrm{cd}_{\mathbb{Z}}(G)=3 \big \}\). \end{enumerate}