Uniform negative immersions and the coherence of one-relator groups

TL;DR

This paper proves that one-relator groups with uniform negative immersions are coherent and satisfy strong subgroup constraints, using linear programming techniques to strengthen previous geometric conditions.

Contribution

The authors establish that uniform negative immersions imply coherence in one-relator groups, answering Baumslag's question and introducing a rationality theorem via linear programming.

Findings

One-relator groups with negative immersions are coherent.

Such groups satisfy the co-Hopf property.

The paper introduces a rationality theorem for uniform negative immersions.

Abstract

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.