Uniform negative immersions and the coherence of one-relator groups

Previously, the authors proved that the presentation complex of a one-relator group G G satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of G 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.