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∈F be a nonprimitive element. The primitivity rank of w, π(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∕⟨⟨w⟩⟩ generated by fewer than π(w) elements is free. In particular, if π(w)>2, then G does not contain any Baumslag–Solitar groups. The hypothesis that π(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 χ(Y)≥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 π(w)>1.