  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--Moldavanskii 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 quasi-convex
one-relator hierarchy if and only if it does not contain a Baumslag--Solitar
subgroup and has a Z-stable one-relator hierarchy.
