Thurston's spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic 3manifolds
Authors:
Feng Luo, Stephan Tillmann and Tian Yang
Journal:
Proc. Amer. Math. Soc. 141 (2013), 335350
MSC (2010):
Primary 57M25, 57N10
Published electronically:
August 17, 2012
MathSciNet review:
2988735
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Abstract: We show that the hyperbolic structure on a closed, orientable, hyperbolic 3manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures and small Seifert fibred structures on closed 3manifolds.
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William P. Thurston: Hyperbolic structures on manifolds I: Deformation of acylindrical manifolds, Annals of Mathematics (2) 124 (1986), no. 2, 203246. MR 855294 (88g:57014)
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William P. Thurston: The geometry and topology of manifolds, Princeton Univ. Math. Dept. (1978). Available from http://msri.org/publications/books/gt3m/.
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Tomoyoshi Yoshida: On ideal points of deformation curves of hyperbolic manifolds with one cusp, Topology 30 (1991), no. 2, 155170. MR 1098911 (92a:57018)
 1.
 Thomas Becker and Volker Weispfenning: Gröbner Bases: A Computational Approach to Commutative Algebra, SpringerVerlag, New York, 1993. MR 1213453 (95e:13018)
 2.
 Riccardo Benedetti and Carlo Petronio: Lectures on Hyperbolic Geometry. SpringerVerlag, Berlin, 1992. MR 1219310 (94e:57015)
 3.
 Benjamin A. Burton: Regina: Normal Surface and Manifold Topology Software, 19992009, http://regina.sourceforge.net/.
 4.
 HuaiDong Cao and XiPing Zhu: A complete proof of the Poincaré and geometrization conjecturesapplication of the HamiltonPerelman theory of the Ricci flow, Asian J. Math. 10 (2006), no. 2, 165492; Erratum, Asian J. Math. 10 (2006), no. 4, p. 663. MR 2233789 (2008d:53090); MR 2282358 (2008d:53091)
 5.
 David Coulson, Oliver A. Goodman, Craig D. Hodgson and Walter D. Neumann: Computing arithmetic invariants of manifolds, Experimental Mathematics 9 (2000), 127152. MR 1758805 (2001c:57014)
 6.
 David Cox, John Little and Donal O'Shea: Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Third edition. Springer, New York, 2007. MR 2290010 (2007h:13036)
 7.
 Nathan M. Dunfield: Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999), no. 3, 623657. MR 1695208 (2000d:57022)
 8.
 Stefano Francaviglia: Hyperbolic volume of representations of fundamental groups of cusped manifolds, IMRN, 2004, no. 9, 425459. MR 2040346 (2004m:57032)
 9.
 Stefano Francaviglia and Ben Klaff: Maximal volume representations are Fuchsian, Geom. Dedicata 117 (2006), 111124. MR 2231161 (2007d:51019)
 10.
 David Gabai, Robert Meyerhoff and Peter Milley: Minimum volume cusped hyperbolic threemanifolds, Journal of the American Mathematical Society 22 (4) (2009), 11571215. MR 2525782 (2011a:57031)
 11.
 William Jaco and Jeffrey Tollefson: Algorithms for the complete decomposition of a closed manifold, Illinois J. Math. 39 (1995), no. 3, 358406. MR 1339832 (97a:57014)
 12.
 Feng Luo: Continuity of the volume of simplices in classical geometry, Commun. Contemp. Math. 8 (2006), no. 3, 411431. MR 2230889 (2007f:52026)
 13.
 Feng Luo: Volume optimization, normal surfaces and Thurston's equation on triangulated manifolds, Preprint, arXiv:0903.1138v1.
 14.
 Feng Luo and Stephan Tillmann: Angle structures and normal surfaces, Trans. Amer. Math. Soc. 360 (2008), no. 6, 28492866. MR 2379778 (2009b:57046)
 15.
 Bruce Kleiner and John Lott: Notes on Perelman's papers, Geom. Topol. 12 (2008), 25872855. MR 2460872 (2010h:53098)
 16.
 Rüdiger Loos: Computing in Algebraic Extensions, from: ``Computer Algebra: Symbolic and Algebraic Computation'' (Bruno Buchberger, George Edwin Collins, Rüdiger Loos, Rudolf Albrecht, editors), SpringerVerlag, Vienna, 1983, 173188. MR 728972
 17.
 Jason Manning: Algorithmic detection and description of hyperbolic structures on closed
manifolds with solvable word problem, Geometry & Topology 6 (2002) 126. MR 1885587 (2002k:57043)
 18.
 Peter Milley: Minimum volume hyperbolic manifolds, Journal of Topology 2 (2009), no. 2, 181192. MR 2499442 (2010d:57018)
 19.
 John Milnor: Collected papers. Vol. 1. Geometry. Publish or Perish, Inc., Houston, TX, 1994. MR 1277810 (95c:01043)
 20.
 John Morgan and Gang Tian: Ricci flow and the Poincaré conjecture, AMS, Providence, 2007. MR 2334563 (2008d:57020)
 21.
 John Morgan and Gang Tian: Completion of the Proof of the Geometrization Conjecture, arXiv:0809.4040v1.
 22.
 Robert Penner: The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), no. 2, 299339. MR 919235 (89h:32044)
 23.
 Grisha Perelman: The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159.
 24.
 Grisha Perelman: Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, arXiv:math/0307245.
 25.
 Grisha Perelman: Ricci flow with surgery on threemanifolds, arXiv:math/0303109.
 26.
 John G. Ratcliffe: Foundations of hyperbolic manifolds. Second edition, Graduate Texts in Mathematics, 149. Springer, New York, 2006. MR 2249478 (2007d:57029)
 27.
 J. Hyam Rubinstein: An algorithm to recognise small Seifert fiber spaces, Turkish J. Math. 28 (2004), 7587. MR 2056761 (2005b:57042)
 28.
 Zlil Sela: The isomorphism problem for hyperbolic groups I, Annals of Mathematics (2) 141 (1995), 217283. MR 1324134 (96b:20049)
 29.
 William P. Thurston: Hyperbolic structures on manifolds I: Deformation of acylindrical manifolds, Annals of Mathematics (2) 124 (1986), no. 2, 203246. MR 855294 (88g:57014)
 30.
 William P. Thurston: The geometry and topology of manifolds, Princeton Univ. Math. Dept. (1978). Available from http://msri.org/publications/books/gt3m/.
 31.
 Tomoyoshi Yoshida: On ideal points of deformation curves of hyperbolic manifolds with one cusp, Topology 30 (1991), no. 2, 155170. MR 1098911 (92a:57018)
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Additional Information
Feng Luo
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email:
fluo@math.rutgers.edu
Stephan Tillmann
Affiliation:
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
Email:
tillmann@maths.uq.edu.au
Tian Yang
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email:
tianyang@math.rutgers.edu
DOI:
http://dx.doi.org/10.1090/S000299392012112201
Keywords:
Hyperbolic 3–manifold,
triangulation,
parameter space,
Thurston’s gluing equations
Received by editor(s):
October 2, 2010
Received by editor(s) in revised form:
April 8, 2011
Published electronically:
August 17, 2012
Additional Notes:
Research of the first and third authors was supported in part by the NSF
Research of the second author was partially funded by a UQ New Staff Research StartUp Fund and under the Australian Research Council’s Discovery funding scheme (DP1095760)
Communicated by:
Daniel Ruberman
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
