Negative immersions for onerelator groups
Abstract
We prove a freeness theorem for lowrank subgroups of onerelator groups. Let $F$ be a free group, and let $w\in F$ be a nonprimitive element. The primitivity rank of $w$, $\pi(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprimitive element. Then any subgroup of the onerelator group $G=F/\langle\langle w\rangle\rangle$ generated by fewer than $\pi(w)$ elements is free. In particular, if $\pi(w)>2$ then $G$ doesn't contain any BaumslagSolitar groups. The hypothesis that $\pi(w)>2$ implies that the presentation complex $X$ of the onerelator group $G$ has negative immersions: if a compact, connected complex $Y$ immerses into $X$ and $\chi(Y)\geq 0$ then $Y$ is Nielsen equivalent to a graph. The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and onerelator groups, including Magnus' Freiheitssatz and theorems of Lyndon, Baumslag, Stallings and DuncanHowie. The dependence theorem strengthens Wise's $w$cycles conjecture, proved independently by the authors and HelferWise, which implies that the onerelator complex $X$ has nonpositive immersions when $\pi(w)>1$.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.02671
 Bibcode:
 2018arXiv180302671L
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Geometric Topology
 EPrint:
 40 pages, 6 figures. Version 2 (and the identical version 3) incorporate referees' comments and corrections. Version 4 only introduces a terminological change: "branched immersions" have been rechristened "branched maps". This is the final version accepted for publication