Thurston’s spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic 3–manifolds

Luo, Feng; Tillmann, Stephan; Yang, Tian

doi:10.1090/S0002-9939-2012-11220-1

Thurston’s spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic 3–manifolds
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by Feng Luo, Stephan Tillmann and Tian Yang

Proc. Amer. Math. Soc. 141 (2013), 335-350

DOI: https://doi.org/10.1090/S0002-9939-2012-11220-1

Published electronically: August 17, 2012

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Abstract:

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3–manifold 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 3–manifolds.

References

Thomas Becker and Volker Weispfenning, Gröbner bases, Graduate Texts in Mathematics, vol. 141, Springer-Verlag, New York, 1993. A computational approach to commutative algebra; In cooperation with Heinz Kredel. MR 1213453, DOI 10.1007/978-1-4612-0913-3
Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992. MR 1219310, DOI 10.1007/978-3-642-58158-8
Benjamin A. Burton: Regina: Normal Surface and $3$-Manifold Topology Software, 1999–2009, http://regina.sourceforge.net/.
Huai-Dong Cao and Xi-Ping Zhu, A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. 10 (2006), no. 2, 165–492. MR 2233789, DOI 10.4310/AJM.2006.v10.n2.a2
David Coulson, Oliver A. Goodman, Craig D. Hodgson, and Walter D. Neumann, Computing arithmetic invariants of 3-manifolds, Experiment. Math. 9 (2000), no. 1, 127–152. MR 1758805
David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. An introduction to computational algebraic geometry and commutative algebra. MR 2290010, DOI 10.1007/978-0-387-35651-8
Nathan M. Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999), no. 3, 623–657. MR 1695208, DOI 10.1007/s002220050321
Stefano Francaviglia, Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, Int. Math. Res. Not. 9 (2004), 425–459. MR 2040346, DOI 10.1155/S1073792804131619
Stefano Francaviglia and Ben Klaff, Maximal volume representations are Fuchsian, Geom. Dedicata 117 (2006), 111–124. MR 2231161, DOI 10.1007/s10711-005-9033-0
David Gabai, Robert Meyerhoff, and Peter Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009), no. 4, 1157–1215. MR 2525782, DOI 10.1090/S0894-0347-09-00639-0
William Jaco and Jeffrey L. Tollefson, Algorithms for the complete decomposition of a closed $3$-manifold, Illinois J. Math. 39 (1995), no. 3, 358–406. MR 1339832
Feng Luo, Continuity of the volume of simplices in classical geometry, Commun. Contemp. Math. 8 (2006), no. 3, 411–431. MR 2230889, DOI 10.1142/S0219199706002179
Feng Luo: Volume optimization, normal surfaces and Thurston’s equation on triangulated $3$-manifolds, Preprint, arXiv:0903.1138v1.
Feng Luo and Stephan Tillmann, Angle structures and normal surfaces, Trans. Amer. Math. Soc. 360 (2008), no. 6, 2849–2866. MR 2379778, DOI 10.1090/S0002-9947-08-04301-8
Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. MR 2460872, DOI 10.2140/gt.2008.12.2587
R. Loos, Computing in algebraic extensions, Computer algebra, Springer, Vienna, 1983, pp. 173–187. MR 728972, DOI 10.1007/978-3-7091-7551-4_{1}2
Jason Manning, Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem, Geom. Topol. 6 (2002), 1–25. MR 1885587, DOI 10.2140/gt.2002.6.1
Peter Milley, Minimum volume hyperbolic 3-manifolds, J. Topol. 2 (2009), no. 1, 181–192. MR 2499442, DOI 10.1112/jtopol/jtp006
John Milnor, Collected papers. Vol. 1, Publish or Perish, Inc., Houston, TX, 1994. Geometry. MR 1277810
John Morgan and Gang Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR 2334563, DOI 10.1305/ndjfl/1193667709
John Morgan and Gang Tian: Completion of the Proof of the Geometrization Conjecture, arXiv:0809.4040v1.
R. C. Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), no. 2, 299–339. MR 919235
Grisha Perelman: The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159.
Grisha Perelman: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245.
Grisha Perelman: Ricci flow with surgery on three-manifolds, arXiv:math/0303109.
John G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR 2249478
J. Hyam Rubinstein, An algorithm to recognise small Seifert fiber spaces, Turkish J. Math. 28 (2004), no. 1, 75–87. MR 2056761
Z. Sela, The isomorphism problem for hyperbolic groups. I, Ann. of Math. (2) 141 (1995), no. 2, 217–283. MR 1324134, DOI 10.2307/2118520
William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI 10.2307/1971277
William P. Thurston: The geometry and topology of $3$–manifolds, Princeton Univ. Math. Dept. (1978). Available from http://msri.org/publications/books/gt3m/.
Tomoyoshi Yoshida, On ideal points of deformation curves of hyperbolic $3$-manifolds with one cusp, Topology 30 (1991), no. 2, 155–170. MR 1098911, DOI 10.1016/0040-9383(91)90003-M