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Strong cylindricality and the monodromy of bundles

Authors: Kazuhiro Ichihara, Tsuyoshi Kobayashi and Yo’av Rieck
Journal: Proc. Amer. Math. Soc. 143 (2015), 3169-3176
MSC (2010): Primary 57M99, 57R22
Published electronically: March 4, 2015
MathSciNet review: 3336641
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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.

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Kazuhiro Ichihara
Affiliation: Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan

Tsuyoshi Kobayashi
Affiliation: Department of Mathematics, Nara Women’s University, Kitauoya Nishimachi, Nara 630-8506, Japan

Yo’av Rieck
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701

Keywords: 3-manifolds, fiber bundles, hyperbolic manifolds, translation distance
Received by editor(s): September 12, 2013
Received by editor(s) in revised form: February 2, 2014
Published electronically: March 4, 2015
Additional Notes: The first author was supported by JSPS KAKENHI Grant Number 23740061.
The second author was supported by JSPS KAKENHI Grant Number 25400091.
This work was partially supported by a grant from the Simons Foundation (Grant Number 283495 to Yo’av Rieck).
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2015 American Mathematical Society