COHERENCE, LOCAL INDICABILITY AND NONPOSITIVE IMMERSIONS

Abstract

We examine 2-complexes $X$ with the property that for any compact connected  $Y$ , and immersion $Y\rightarrow X$ , either $\unicode[STIX]{x1D712}(Y)\leqslant 0$ or $\unicode[STIX]{x1D70B}_{1}Y=1$ . 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.