Recalibrating $\mathbb{R}$-order trees and $\mbox{Homeo}_+(S^1)$-representations of link groups

Abstract

In this paper we study the left-orderability of $3$-manifold groups using an enhancement, called recalibration, of Calegari and Dunfield's "flipping" construction, used for modifying $\mbox{Homeo}_+(S^1)$-representations of the fundamental groups of closed $3$-manifolds. The added flexibility accorded by recalibration allows us to produce $\mbox{Homeo}_+(S^1)$-representations of hyperbolic link exteriors so that a chosen element in the peripheral subgroup is sent to any given rational rotation. We apply these representations to show that the branched covers of families of links associated to arbitrary epimorphisms of the link group onto a finite cyclic group are left-orderable. This applies, for instance, to fibered hyperbolic strongly quasipositive links. Our result on the orderability of branched covers implies that the degeneracy locus of any pseudo-Anosov flow on an alternating knot complement must be meridional, which generalizes the known result that the fractional Dehn twist coefficient of any hyperbolic fibered alternating knot is zero. Applications of these representations to order-detection of slopes are also discussed in the paper.