  Previously, the authors proved that the presentation complex of a one-relator
group $G$ satisfies a geometric condition called negative immersions if every
two-generator, one-relator subgroup of $G$ is free. Here, we prove that
one-relator groups with negative immersions are coherent, answering a question
of Baumslag in this case. Other strong constraints on the finitely generated
subgroups also follow such as, for example, the co-Hopf property. The main new
theorem strengthens negative immersions to uniform negative immersions, using a
rationality theorem proved with linear-programming techniques.
