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 h ∈ H, x ∈ 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 B that does not lie in A B is Cohen-Lyndon aspherical in A 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.