Negative immersions for one-relator groups

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be free group, and let w∈F nonprimitive element. The primitivity rank w, π(w), is the smallest subgroup containing w as an imprimitive Then any group G=F∕⟨⟨w⟩⟩ generated by fewer than π(w) elements free. In particular, if π(w)>2, then G does not contain Baumslag–Solitar hypothesis that π(w)>2 implies presentation complex X has negative immersions: compact, connected Y immerses into χ(Y)≥0, Nielsen reduces to graph. consequence dependence groups, which several classical facts about including Magnus’ Freiheitssatz theorems Lyndon, Baumslag, Stallings, Duncan–Howie. strengthens Wise’s w-cycles conjecture, proved independently authors Helfer–Wise, nonpositive immersions when π(w)>1.