Bowditch Taut Spectrum and Dimensions of Groups

Abstract

For a finitely generated group $G$, let $H(G)$ denote Bowditch's taut loop length spectrum. We prove that if $G=(A\ast B) / \langle\!\langle \mathcal R \rangle\!\rangle $ is a $C'(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $H(G)$ is equivalent to $H(A) \cup H(B)$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch's construction of continuously many non-quasi-isometric $C'(1/6)$ small cancellation $2$-generated groups to obtain our main result: Let $\mathcal{G}$ denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups: $\bullet\left\{G\in \mathcal{G} \colon \underline{\mathrm{cd}}(G) = 2 \text{ and } \underline{\mathrm{gd}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \underline{\underline{\mathrm{cd}}}(G) = 2 \text{ and } \underline{\underline{\mathrm{gd}}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \mathrm{cd}_{\mathbb{Q}}(G)=2 \text{ and } \mathrm{cd}_{\mathbb{Z}}(G)=3 \right\}$ On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented $C'(1/12)$ small cancellation quotients of free products, in particular, this produces new examples of groups exhibiting an Eilenberg-Ganea phenomenon for families. We also show that if there is a finitely presented counter-example to the Eilenberg-Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counter-examples.