Negative immersions for one-relator groups

Abstract

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be a free group, and let $w\in F$ be a nonprimitive element. The primitivity rank of w, $\mathrm{\pi }(w)$, is the smallest rank of a subgroup of F containing w as an imprimitive element. Then any subgroup of the one-relator group $G=F∕\mathrm{⟨}\mathrm{⟨}w\mathrm{⟩}\mathrm{⟩}$ generated by fewer than $\mathrm{\pi }(w)$ elements is free. In particular, if $\mathrm{\pi }(w)>2$, then G does not contain any Baumslag–Solitar groups.

The hypothesis that $\mathrm{\pi }(w)>2$ implies that the presentation complex X of the one-relator group G has negative immersions: if a compact, connected complex Y immerses into X and $\mathit{\chi }(Y)\ge 0$, then Y Nielsen reduces to a graph.

The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and one-relator groups, including Magnus’ Freiheitssatz and theorems of Lyndon, Baumslag, Stallings, and Duncan–Howie.

The dependence theorem strengthens Wise’s w-cycles conjecture, proved independently by the authors and Helfer–Wise, which implies that the one-relator complex X has nonpositive immersions when $\mathrm{\pi }(w)>1$.