COHERENCE, LOCAL INDICABILITY AND NONPOSITIVE IMMERSIONS

We examine 2-complexes X with the property that for any compact connected Y, and immersion Y → X, either χ(Y) ≤ 0 or π1Y = 1. The mapping torus of an endomorphism of a free group has this property. Every irreducible 3-manifold with boundary has a spine with this property. We show that the fundamental group of any 2-complex with this property is locally indicable. We outline evidence supporting the conjecture that this property implies coherence. We connect the property to asphericity. Finally, we prove coherence for 2-complexes with a stricter form of this property. As a corollary, every one-relator group with torsion is coherent.