Orderability and dehn filling

Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3-spheres. Specifically, for a compact 3-manifold M with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of M have left-orderable fundamental groups. Our technique uses certain representations from π₁(M) into PSL₂ℝ, which we organize into an infinite graph in H¹(∂M;ℝ) called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.