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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Thurston's spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic 3-manifolds


Authors: Feng Luo, Stephan Tillmann and Tian Yang
Journal: Proc. Amer. Math. Soc. 141 (2013), 335-350
MSC (2010): Primary 57M25, 57N10
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.


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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/S0002-9939-2012-11220-1
PII: S 0002-9939(2012)11220-1
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 Start-Up 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.