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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Construction and recognition of hyperbolic 3-manifolds with geodesic boundary


Authors: Roberto Frigerio and Carlo Petronio
Journal: Trans. Amer. Math. Soc. 356 (2004), 3243-3282
MSC (2000): Primary 57M50; Secondary 57M25
Published electronically: August 26, 2003
MathSciNet review: 2052949
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Abstract: We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical decomposition, to an algorithm that computes the Kojima decomposition.

Our theory has been exploited to classify all the orientable finite-volume hyperbolic $3$-manifolds with non-empty compact geodesic boundary admitting an ideal triangulation with at most four tetrahedra. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boundary in which the boundary is non-compact. We include this case using a suitable adjustment of the notion of ideal triangulation, and we show how this case arises within the theory of knots and links.


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Additional Information

Roberto Frigerio
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
Email: frigerio@sns.it

Carlo Petronio
Affiliation: Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno Pisano, 25/B, 6126 Pisa, Italy
Email: petronio@dma.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03378-6
PII: S 0002-9947(03)03378-6
Received by editor(s): December 1, 2001
Received by editor(s) in revised form: March 20, 2003
Published electronically: August 26, 2003
Article copyright: © Copyright 2003 American Mathematical Society