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

MathSciNet review:
2988735

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Abstract | References | Similar Articles | Additional Information

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

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.