Orderability and Dehn filling

Abstract

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 ${\pi }_1\left(M\right)$ into $\widetilde{{PSL}_2ℝ}$, which we organize into an infinite graph in $H^1\left(\partial M;ℝ\right)$ 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.