On the local-indicability Cohen-Lyndon Theorem

Details On the local-indicability Cohen-Lyndon Theorem On the local-indicability Cohen-Lyndon Theorem

2010-02-09

arxiv.org/abs/1002.1934v1

Glasgow Mathematical Journal 53 (2011), 637--656

Mathematics

Group Theory

15 pages, no figures

Scientific paper

10.1017/S0017089511000231

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.

Affiliated with

Also associated with

No associations

Rating

On the local-indicability Cohen-Lyndon Theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the local-indicability Cohen-Lyndon Theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the local-indicability Cohen-Lyndon Theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-125240