One-relator hierarchies

We prove that one-relator groups with negative immersions are hyperbolic and virtually special; this resolves a recent conjecture of Louder and Wilton. As a consequence, one-relator groups with negative immersions are residually finite, linear, and have isomorphism problem decidable among one-relator groups. Using the fact that parafree one-relator groups have negative immersions, we answer a question of Baumslag’s from 1986. The main new tool we develop is a refinement of the classic Magnus–Moldavanskiĭ hierarchy for one-relator groups. We introduce the notions of Z-stable HNN-extensions and Z-stable hierarchies. We then show that a one-relator group is hyperbolic and has a quasiconvex one-relator hierarchy if and only if it does not contain a Baumslag–Solitar subgroup and has a Z-stable one-relator hierarchy.