Order-detection of slopes on the boundaries of knot manifolds

TL;DR

This paper explores how various notions of order-detection on knot manifolds relate to the left-orderability of fundamental groups of 3-manifolds obtained via Dehn filling, connecting Heegaard-Floer homology, foliations, and group actions.

Contribution

It introduces new notions of order-detection for slopes on knot manifolds and links these to existing theories, proposing conjectures on the structure of left-orderable 3-manifold groups.

Findings

Parallel results with Heegaard-Floer and foliation slope detection

Conjectured links between Floer homology and boundary foliations

Dynamical constraints on group actions for Floer simple 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 existing results in the case of Heegaard-Floer slope detection and 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 plays 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.