| Summary: | 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, Cohen and Lyndon (D. E. Cohen and R. C. Lyndon, Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963), 526-537) independently showed that in each free group each non-trivial element is Cohen-Lyndon aspherical. Their proof used the celebrated induction method devised by Magnus in 1930 to study one-relator groups. In 1987, Edjvet and Howie (M. Edjvet and J. Howie, A Cohen-Lyndon theorem for free products of locally indicable groups, J. Pure Appl. Algebra 45 (1987), 41-44) 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. Their proof used the original Cohen-Lyndon theorem. Using Bass-Serre theory, the original Cohen-Lyndon theorem 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 stabilisers, then each element of F that fixes no vertex of T is Cohen-Lyndon aspherical in F. Conversely, by Bass-Serre theory, the original Cohen-Lyndon theorem and the Edjvet-Howie theorem are immediate consequences of the local-indicability Cohen-Lyndon theorem. In this paper we give a detailed review of a Bass-Serre theoretical form of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability Cohen-Lyndon theorem that uses neither Magnus induction nor the original Cohen-Lyndon theorem. We conclude with a review of some standard applications of Cohen-Lyndon asphericity
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