Strong cylindricality and the monodromy of bundles

TL;DR

This paper proves that in certain 3-manifolds, surfaces with sufficiently high genus relative to the triangulation are strongly cylindrical, and provides an alternative proof for finiteness of certain fibrations in hyperbolic 3-manifolds.

Contribution

It establishes a genus bound ensuring strong cylindricality of surfaces in 3-manifolds and offers a new proof for finiteness of specific fibrations in hyperbolic 3-manifolds.

Findings

Surfaces with genus at least 38 times the number of tetrahedra are strongly cylindrical.

Every closed hyperbolic 3-manifold has finitely many fibrations over the circle with connected fiber and translation distance not one.

Provides an alternative proof for a known finiteness result in hyperbolic 3-manifolds.

Abstract

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, $(A, \partial A)$ 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 $38 t$, 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.