关于局部可导性的Cohen-Lyndon定理
On the local-indicability Cohen-Lyndon Theorem
摘要：
对于群$H$和$H$的子集$X$，我们用${}^HX$表示集合$\{hxh^{-1} \mid h \in H, x \in X\}$，当$X$是$H$的自由生成集时，我们说集合${}^HX$是$H$的Whitehead子集。对于群$F$和$F$的元素$r$，如果${}^F\{r\}$是由${}^F\{r\}$生成的$F$的子群的Whitehead子集，则我们说$r$在$F$中是Cohen-Lyndon非球面的。1963年，D. E. Cohen和R. C.林登独立地表明，在每个自由群中，每个非平凡元素都是科恩-林登非球面的。1987年，M. Edjvet和J. Howie证明了，如果$A$和$B$是局部可指示群，则$A \ast B$的每个不位于$A \cup B$中的循环约化元素在$A \ast B$中是Cohen-Lyndon非球面的。利用Bass-Serre理论和Edjvet-Howie定理，可以推导出局部可标性Cohen-Lyndon定理：如果$F$是一个局部可标群，$T$是一个具有平凡边稳定子的$F$ -树，则$F$中每个不固定$T$顶点的元素在$F$中是Cohen-Lyndon非球面的.相反，科恩-林登定理和埃德维特-豪伊定理是局部可导性科恩-林登定理的直接结果。在这篇文章中，我们对Howie归纳法进行了详细的回顾，并将Edjvet和Howie的论点整理成局部可导性Cohen-Lyndon定理的Howie归纳证明，该证明不使用Magnus归纳法或Cohen-Lyndon定理。最后，我们的一些标准的应用程序的科恩-林登非球面性的审查。
英文摘要：
For a group $H$ and a subset $X$ of $H$, we let ${}^HX$ denote the set $\{hxh^{-1} \mid h \in H, x \in X\}$, and when $X$ is a free-generating set of $H$, we say that the set ${}^HX$ is a Whitehead subset of $H$. For a group $F$ and an element $r$ of $F$, we say that $r$ is Cohen-Lyndon aspherical in $F$ if ${}^F\{r\}$ is a Whitehead subset of the subgroup of $F$ that is generated by ${}^F\{r\}$. In 1963, D. E. Cohen and R. C. Lyndon independently showed that in each free group each non-trivial element is Cohen-Lyndon aspherical. In 1987, M. Edjvet and J. Howie showed that if $A$ and $B$ are locally indicable groups, then each cyclically reduced element of $A \ast B$ that does not lie in $A \cup B$ is Cohen-Lyndon aspherical in $A \ast B$. Using Bass-Serre Theory and the Edjvet-Howie Theorem, one can deduce the local-indicability Cohen-Lyndon Theorem: if $F$ is a locally indicable group and $T$ is an $F$-tree with trivial edge stabilizers, then each element of $F$ that fixes no vertex of $T$ is Cohen-Lyndon aspherical in $F$. Conversely, the Cohen-Lyndon Theorem and the Edjvet-Howie Theorem are immediate consequences of the local-indicability Cohen-Lyndon Theorem. In this article, we give a detailed review of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability Cohen-Lyndon Theorem that does not use Magnus induction or the Cohen-Lyndon Theorem. We conclude with a review of some standard applications of Cohen-Lyndon asphericity.
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