群的Bowditch Taut谱与维数
Bowditch Taut Spectrum and dimensions of groups
摘要：
对于一个随机生成的群$G$，令$H(G)$表示Bowditch的紧环长度谱。证明了如果$G=(A\ast B) / \langle\!\langle \mathcal R \rangle\!\rangle $是一个群的自由积的$C'(1/12)$小消去商，则$H(G)$等价于$H(A) \cup H(B)$ .我们使用这一结果连同界限上同调和几何尺寸，以及Bowditch的建设连续许多非准等距$C'(1/6)$小取消$2$生成的群体，以获得我们的主要结果：让$\mathcal{G}$表示类的$\mathcal{G}$生成的群体。以下子类包含连续许多单端非准等距群：$\bullet\left\{G\in \mathcal{G} \colon \underline{\mathrm{cd}}(G) = 2 \text{ and } \underline{\mathrm{gd}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \underline{\underline{\mathrm{cd}}}(G) = 2 \text{ and } \underline{\underline{\mathrm{gd}}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \mathrm{cd}_{\mathbb{Q}}(G)=2 \text{ and } \mathrm{cd}_{\mathbb{Z}}(G)=3 \right\}$在我们的方式来证明上述结果，我们表明，上述定义的类是封闭的下采取相对较少的免费产品的$C'(1/12)$小取消代理人，特别是，这产生新的例子的群体表现出一个Eilenberg-Ganea现象的家庭。我们还证明了，如果有一个Eilenberg-Ganea猜想的反例，那么有连续的许多Eilenberg-Ganea猜想生成的单端非拟等距反例。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
