  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.
