Left orderability and Anosov flows

Let $M$ be a connected, closed, orientable, irreducible $3$-manifold. We show that: if $M$ admits a co-orientable taut foliation which is the stable or unstable foliation of an Anosov flow, then $\pi_1(M)$ is left orderable. In addition, if $M$ admits an Anosov flow, then either $\pi_1(M)$ is left orderable or $\pi_1(M)$ has an index $2$ left orderable subgroup. Combined with a theorem of Calegari, if $M$ is hyperbolic and admits a co-orientable taut foliation $\mathcal{F}$, then either $\pi_1(M)$ is left orderable, or there are a pair of very full genuine laminations transverse to $\mathcal{F}$.Comment: 14 pages, 13 figures. Comments are welcome