Taut foliations in branched cyclic covers and left-orderable groups

Abstract

We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology 3-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible 3-manifolds containing a genus 1 fibred knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibred knot in an integer homology 3-sphere admits a co-oriented taut foliation and has left-orderable fundamental group, even if the surgered manifold does not, and that the same holds for many branched covers of satellite knots with braided patterns. A key fact used in our proofs is that the Euler class of a universal circle representation associated to a co-oriented taut foliation coincides with the Euler class of the foliation's tangent bundle. Though known to experts, no proof of this important result has appeared in the literature. We provide such a proof in the paper.Conjecture 1.1 (Conjecture 1 in [BGW], Conjecture 5 in [Ju]). Assume that M is a closed, connected, irreducible, orientable 3-manifold. Then the following statements are equivalent.(1) M is not a Heegaard Floer L-space,(2) M admits a co-orientable taut foliation,(3) π 1 (M ) is left-orderable.

Branched covers of quasi‐positive links and L‐spaces

“…It was observed in [, Theorem 4.1] that if the fractional Dehn twist coefficient of is at least 1 in absolute value, then work of Roberts as interpreted in implies that is CTF. In a recent preprint , Boyer and Hu show that the same condition implies that is LO. Since the fractional Dehn twist coefficient of is times the fractional Dehn twist of , and since the fractional Dehn twist coefficient of a hyperbolic fibred strongly quasi‐positive knot is non‐zero, is CTF and LO for large .…”

Left-orderability, branched covers and double twist knots

“…Let Σ n (L) denote the n-fold cyclic branched cover of the 3-sphere branched over a link L. Such manifolds have provided a rich family on which to investigate the L-space conjecture. The conjecture has been confirmed for Σ 2 (L) when L is a non-split alternating link [OS05,BGW13] or the closure of a 3-braid [Bal08,BH19,LW14].…”

Promoting circular-orderability to left-orderability

“…At present, almost all approaches to left ordering π 1 (M ) involve first finding a faithful action of π 1 (M ) on S 1 , either via representations of π 1 (M ) into PSL(2, R) (which in turn acts on S 1 ), or by creating a co-orientable, taut foliation of M and then applying Thurston's universal circle construction to arrive at ρ : π 1 (M ) → Homeo + (S 1 ). For examples of these approaches, see [4,16]. In either case the construction of the representation or foliation must be done with great care, so as to guarantee that the corresponding action on S 1 has trivial Euler class-thus guaranteeing that π 1 (M ) is left-orderable (cf.…”