On the local-indicability Cohen-Lyndon Theorem

Abstract

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.