arXiv 2608.18929math.GR

关于群表示的\$\boldsymbol{u}\$-替换

On $\textbf{u}$-substitutions for group presentations

Kirk Mcdermott

首次发表 2026-08-20

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中文总结 AI 辅助

本文研究群表示的\$\boldsymbol{u}\$-替换，综述相关结果并记录新结构性质，将平凡群表示代入\$\mathbb{Z}\$的亏格1群表示得到的群类\$\mathcal{G}(\mathcal{B})\$是2-纽结群的真子集且包含所有经典纽结群。

中文摘要 AI 辅助

我们研究由表示\$\mathcal{P}=\langle \boldsymbol{x}: \boldsymbol{r} \rangle\$给出的群\$\hat{G}\$，其中定义关系\$\boldsymbol{r} \subseteq F(\boldsymbol{x})\$由\$F(\boldsymbol{x})\$中的一组子字构成，即\$\boldsymbol{r}\$允许\$\boldsymbol{u}\$-替换，具体指存在同态\$\epsilon: F(\boldsymbol{u}) \rightarrow F(\boldsymbol{x})\$和子集\$\mathbf{v} \subseteq F(\boldsymbol{u})\$使得\$\boldsymbol{r}=\epsilon(\mathbf{v})\$。等价地，\$\mathcal{P}= \langle \boldsymbol{x}: \epsilon (\mathbf{v}) \rangle\$被称为群\$G\$的表示\$\mathcal{G}= \langle \boldsymbol{u}: \mathbf{v} \rangle\$与群\$H\$的表示\$\mathcal{H}= \langle \boldsymbol{x}: \epsilon(\boldsymbol{u}) \rangle\$的复合。我们综述已知结果并记录文献中未明确出现的结构性质，例如存在\$\hat{G}\$的相对表示\$\langle G, \boldsymbol{x}: \boldsymbol{u}= \epsilon(\boldsymbol{u}) \rangle\$，因此存在自然同态\$\epsilon: G \rightarrow \hat{G}\$，可研究其相关问题（如单射性、有限性）。作为应用，我们研究从平凡群的表示代入\$\mathbb{Z}\$的亏格1群表示得到的群类\$\mathcal{G}(\mathcal{B})\$，结果表明\$\mathcal{G}(\mathcal{B})\$是2-纽结群类的真子集，且真包含所有经典纽结群；还研究了包含三叶结群的子族，该子族使用平凡群的Higman表示。

英文摘要

We investigate groups $\hat{G}$ given by a presentation $\mathcal{P}=\langle \textbf{x}: \textbf{r} \rangle$ whose relators $\textbf{r} \subseteq F(\textbf{x})$ are comprised of a set of subwords in $F(\textbf{x})$, i.e. $\textbf{r}$ admits a $\textbf{u}$-substitution in the sense that there exists a homomorphsim $ε: F(\textbf{u}) \rightarrow F(\textbf{x})$ and a subset $\mathbf{v} \subseteq F(\textbf{u})$ such that $\textbf{r}=ε(\textbf{v})$. Equivalently, $\mathcal{P}= \langle \textbf{x}: ε(\textbf{v}) \rangle$ is referred to as the composition of the presentation $\mathcal{G}= \langle \textbf{u}: \textbf{v} \rangle$ with $\mathcal{H}= \langle \textbf{x}: ε(\textbf{u}) \rangle$ of the groups $G$ and $H$, respectively. We survey known results and record structural properties which do not explicitly appear in the literature, e.g. that there is the relative presentation $\langle G, \textbf{x}: \textbf{u}= ε(\textbf{u}) \rangle$ for $\hat{G}$. Thus there is a natural map $ε: G \rightarrow \hat{G}$ and we may consider the associated problems (e.g. injectivity, finiteness). As an application, we investigate the class of groups $\mathcal{G}(\mathcal{B})$ obtained from substituting a presentation of the trivial group into a deficiency one group presentation for $\mathbb{Z}$. Our results show $\mathcal{G}(\mathcal{B})$ is a proper subset of the class of 2-knot groups and properly contains all classical knot groups. A subfamily is investigated which includes the group of the trefoil knot and uses Higman's presentations for the trivial group.