Recalibrating R $\mathbb {R}$ -order trees and Homeo + ( S 1 ) $\mbox{Homeo}_+(S^1)$ -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 ${\text{Homeo}}_{+}\left(S^1\right)$-representations of the fundamental groups of closed 3-manifolds. The added flexibility accorded by recalibration allows us to produce ${\text{Homeo}}_{+}\left(S^1\right)$-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 epimorphisms of the link group onto a finite cyclic group are left-orderable. This applies, for instance, to fibred hyperbolic strongly quasi-positive 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 generalises the known result that the fractional Dehn twist coefficient of any hyperbolic fibred alternating knot is zero. Applications of these representations to order detection of slopes are also discussed in the paper.