Order-detection of slopes on the boundaries of knot manifolds

Abstract

Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres, obtained by gluing compact manifolds along torus boundary components, have left-orderable fundamental groups and when a Dehn filling of a knot manifold has a left-orderable fundamental group. Our developments parallel the results by Hanselman et al. (2020) in the case of Heegaard Floer slope detection and by Boyer et al. (2021) in the case of foliation slope detection, leading to several conjectured structure theorems that connect relative Heegaard Floer homology and the boundary behaviour of co-oriented taut foliations with the set of left-orders supported by the fundamental group of a 3-manifold. The dynamics of the actions of 3-manifold groups on the real line play a key role in our constructions and proofs. Our analysis leads to conjectured dynamical constraints on such actions in the case where the underlying manifold is Floer simple.