Proceedings of the American Mathematical Society

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

 

 

Geodesic ideal triangulations exist virtually


Authors: Feng Luo, Saul Schleimer and Stephan Tillmann
Journal: Proc. Amer. Math. Soc. 136 (2008), 2625-2630
MSC (2000): Primary 57N10, 57N15; Secondary 20H10, 22E40, 51M10
Published electronically: March 10, 2008
MathSciNet review: 2390535
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups.


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

  • 1. Young-Eun Choi, Positively oriented ideal triangulations on hyperbolic three-manifolds, Topology 43 (2004), no. 6, 1345–1371. MR 2081429, 10.1016/j.top.2004.02.002
  • 2. D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988), no. 1, 67–80. MR 918457
  • 3. Roberto Frigerio, On deformations of hyperbolic 3-manifolds with geodesic boundary, Algebr. Geom. Topol. 6 (2006), 435–457 (electronic). MR 2220684, 10.2140/agt.2006.6.435
  • 4. J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0248844
  • 5. Sadayoshi Kojima, Polyhedral decomposition of hyperbolic 3-manifolds with totally geodesic boundary, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math., vol. 20, Kinokuniya, Tokyo, 1992, pp. 93–112. MR 1208308
  • 6. Sadayoshi Kojima: Polyhedral Decomposition of Hyperbolic Manifolds with Boundary, On the geometric structure of manifolds, edited by Dong Pyo Chi, Proc. of Workshop in Pure Math., 10, part III, 37-57 (1990).
  • 7. Carl W. Lee, Subdivisions and triangulations of polytopes, Handbook of discrete and computational geometry, CRC Press Ser. Discrete Math. Appl., CRC, Boca Raton, FL, 1997, pp. 271–290. MR 1730170
  • 8. D. D. Long and G. A. Niblo, Subgroup separability and 3-manifold groups, Math. Z. 207 (1991), no. 2, 209–215. MR 1109662, 10.1007/BF02571384
  • 9. A. Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940), 405–422 (Russian, with English summary). MR 0003420
  • 10. Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, 10.1016/0040-9383(85)90004-7
  • 11. Carlo Petronio and Joan Porti, Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem, Expo. Math. 18 (2000), no. 1, 1–35. MR 1751141
  • 12. John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730
  • 13. William P. Thurston: The geometry and topology of $ 3$-manifolds, Princeton Univ. Math. Dept. (1978). Available from http://msri.org/publications/books/gt3m/.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57N10, 57N15, 20H10, 22E40, 51M10

Retrieve articles in all journals with MSC (2000): 57N10, 57N15, 20H10, 22E40, 51M10


Additional Information

Feng Luo
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email: fluo@math.rutgers.edu

Saul Schleimer
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email: saulsch@math.rutgers.edu

Stephan Tillmann
Affiliation: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia
Email: tillmann@ms.unimelb.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09387-8
Keywords: Hyperbolic manifold, ideal triangulation, partially truncated triangulation, subgroup separability
Received by editor(s): April 9, 2007
Published electronically: March 10, 2008
Additional Notes: The research of the first author was supported in part by the NSF
The second author was partly supported by the NSF (DMS-0508971).
The third author was supported under the Australian Research Council’s Discovery funding scheme (project number DP0664276).
This work is in the public domain.
Communicated by: Daniel Ruberman