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 | References | Similar Articles | Additional Information

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 -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:
https://doi.org/10.1090/S0002-9947-03-03378-6

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