  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 non-primitive 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 one-relator
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
Baumslag--Solitar groups.
  The hypothesis that $\pi(w)>2$ 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 $\chi(Y)\geq 0$ 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 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 non-positive immersions when
$\pi(w)>1$.
