Strong cylindricality and the monodromy of bundles

A surface $F$ in a 3-manifold $M$ is called cylindrical if $M$ cut open along $F$ admits an essential annulus $A$. If, in addition, $(A,\mathrm{\partial }A)$ is embedded in $(M,F)$, then we say that $F$ is strongly cylindrical. Let $M$ be a connected 3-manifold that admits a triangulation using $t$ tetrahedra and $F$ a two-sided connected essential closed surface of genus $g(F)$. We show that if $g(F)$ is at least $38t$, then $F$ is strongly cylindrical. As a corollary, we give an alternative proof of the assertion that every closed hyperbolic 3-manifold admits only finitely many fibrations over the circle with connected fiber whose translation distance is not one, which was originally proved by Saul Schleimer.