Bowditch Taut Spectrum and Dimensions of Groups

Abstract

For a finitely generated group G, let $\mathit{H}(\mathit{G})$ denote Bowditch’s taut loop length spectrum. We prove that if $\mathit{G}=(\mathit{A}\ast \mathit{B})/⟨⟨\mathcal{R}⟩⟩$ is a ${\mathit{C}}^{\prime }(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $\mathit{H}(\mathit{G})$ is equivalent to $\mathit{H}(\mathit{A})\cup \mathit{H}(\mathit{B})$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch’s construction of continuously many non-quasi-isometric ${\mathit{C}}^{\prime }(1/6)$ 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:

(1) $\{\mathit{G}\in \mathcal{G}:\underset{\_}{\mathrm{cd}}(\mathit{G})=2\text{and}\underset{\_}{\mathrm{gd}}(\mathit{G})=3\}$;

(2) $\{\mathit{G}\in \mathcal{G}:\underset{\_}{\underset{\_}{\mathrm{cd}}}(\mathit{G})=2\text{and}\underset{\_}{\underset{\_}{\mathrm{gd}}}(\mathit{G})=3\}$;

(3) $\{\mathit{G}\in \mathcal{G}:{\mathrm{cd}}_{\mathbb{Q}}(\mathit{G})=2\text{and}{\mathrm{cd}}_{\mathbb{Z}}(\mathit{G})=3\}$.

On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented ${\mathit{C}}^{\prime }(1/12)$ 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 counterexample to the Eilenberg–Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counterexamples.