Constructing depth one laminations transverse to pseudo-Anosov flows

Abstract

Given a pseudo-Anosov flow $\ddot{I}•$ on a closed atoroidal $3$--manifold $M$ and a closed surface $S$ almost transverse to $\ddot{I}•$, we give a homological characterization of when $S$ can be completed to an almost transverse depth one lamination or foliation whose set of compact leaves is $S$. As a consequence, we show that the cone of classes in $H^1(M\backslash \backslash S)$ that are positive on the closed orbits of $\ddot{I}•$, when nonempty, is an entire foliation cone of $M\backslash \backslash S$.

30 pages, 14 figures