Incompressible tori transverse to Anosov flows in 3-manifolds

Abstract

We consider Anosov flows in 3-manifolds. Suppose that there is a rank-two free abelian subgroup of the fundamental group of the manifold, so that none of its elements can be represented by a closed orbit of the flow. We then show that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism. As a consequence we prove that if $T$ is an incompressible torus so that no loop in $T$ is freely homotopic to a closed orbit of the flow, then $T$ is isotopic to a transverse torus. Finally, we show that if $T$ is an incompressible torus transverse to the stable foliation, then either there is a closed leaf in the induced foliation in $T$, or the flow is topologically conjugate to a suspension Anosov flow.