Left orderability and taut foliations with orderable cataclysm

TL;DR

This paper proves that certain 3-manifolds with specific taut foliations have left orderable fundamental groups, providing an elementary approach and extending to many Dehn fillings of manifolds with pseudo-Anosov flows.

Contribution

It establishes a new link between taut foliations with orderable cataclysm and left orderability of the fundamental group, avoiding Thurston's universal circle action.

Findings

If a 3-manifold admits a taut foliation with orderable cataclysm, its fundamental group is left orderable.

The result applies to manifolds with Anosov flows with co-orientable stable foliations, without using Thurston's universal circle.

It extends to infinitely many Dehn fillings of manifolds with pseudo-Anosov flows with co-orientable stable foliations.

Abstract

Let $M$ be a connected, closed, orientable, irreducible $3$-manifold. We show that: if $M$ admits a co-orientable taut foliation $\mathcal{F}$ with orderable cataclysm, then $\pi_1(M)$ is left orderable. This provides an elementary proof that $\pi_1(M)$ is left orderable if $M$ admits an Anosov flow with a co-orientable stable foliation without using Thurston's universal circle action. Furthermore, for every closed orientable 3-manifold that admits a pseudo-Anosov flow $X$ with a co-orientable stable foliation, our result applies to infinitely many of Dehn fillings along the union of singular orbits of $X$.