Left orderability and taut foliations with orderable cataclysm

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 $\ddot{I}{€}_1(M)$ is left orderable. This provides an elementary proof that $\ddot{I}{€}_1(M)$ 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 $X$.

16 pages, 13 figures; v3: accepted version