Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2015-12473-2
Published electronically: March 4, 2015
MathSciNet review: 3336641
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045-1087. With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553
  • [2] David Bachman and Saul Schleimer, Surface bundles versus Heegaard splittings, Comm. Anal. Geom. 13 (2005), no. 5, 903-928. MR 2216145 (2006m:57027)
  • [3] Mario Eudave-Muñoz and Max Neumann-Coto, Acylindrical surfaces in 3-manifolds and knot complements, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. Special Issue, 147-169. MR 2199345 (2007f:57040)
  • [4] Joel Hass, Acylindrical surfaces in $ 3$-manifolds, Michigan Math. J. 42 (1995), no. 2, 357-365. MR 1342495 (96c:57031), https://doi.org/10.1307/mmj/1029005233
  • [5] William Jaco and Ulrich Oertel, An algorithm to decide if a $ 3$-manifold is a Haken manifold, Topology 23 (1984), no. 2, 195-209. MR 744850 (85j:57014), https://doi.org/10.1016/0040-9383(84)90039-9
  • [6] Tsuyoshi Kobayashi and Yo'av Rieck, A linear bound on the tetrahedral number of manifolds of bounded volume (after Jørgensen and Thurston), Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 27-42. MR 2866921, https://doi.org/10.1090/conm/560/11089
  • [7] Tsuyoshi Kobayashi and Yo'av Rieck, Hyperbolic volume and Heegaard distance, Comm. Anal. Geom. 22 (2014), no. 2, 247-268. MR 3210755, https://doi.org/10.4310/CAG.2014.v22.n2.a3
  • [8] Saul Schleimer, Strongly irreducible surface automorphisms, Topology and geometry of manifolds (Athens, GA, 2001) Proc. Sympos. Pure Math., vol. 71, Amer. Math. Soc., Providence, RI, 2003, pp. 287-296. MR 2024639 (2004j:57027)
  • [9] Z. Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997), no. 3, 527-565. MR 1465334 (98m:20045), https://doi.org/10.1007/s002220050172
  • [10] Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, Available at http://www.math.mcgill.ca/wise/papers.html.
  • [11] Daniel T. Wise, From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, vol. 117, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2012. MR 2986461

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57M99, 57R22

Retrieve articles in all journals with MSC (2010): 57M99, 57R22


Additional Information

Kazuhiro Ichihara
Affiliation: Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan
Email: ichihara@math.chs.nihon-u.ac.jp

Tsuyoshi Kobayashi
Affiliation: Department of Mathematics, Nara Women’s University, Kitauoya Nishimachi, Nara 630-8506, Japan
Email: tsuyoshi@cc.nara-wu.ac.jp

Yo’av Rieck
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: yoav@uark.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12473-2
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

American Mathematical Society