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Construction and recognition of hyperbolic 3-manifolds with geodesic boundary

Author(s): Roberto Frigerio; Carlo Petronio
Journal: Trans. Amer. Math. Soc. 356 (2004), 3243-3282.
MSC (2000): Primary 57M50; Secondary 57M25
Posted: August 26, 2003
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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.

References:

1.
G. AMENDOLA, A calculus for ideal triangulations of three-manifolds with embedded arcs, math.GT/0301219.

2.
S. BASEILHAC, R. BENEDETTI, Quantum hyperbolic state sum invariants of $3$-manifolds, math.GT/0101234.

3.
A. F. BEARDON, ``The geometry of discrete groups'', Graduate Texts in Mathematics, Vol. 91, Springer-Verlag, New York, 1995. MR 85d:22026

4.
R. BENEDETTI, C. PETRONIO, ``Lectures in Hyperbolic Geometry'', Universitext, Springer-Verlag, Berlin, 1992. MR 94e:57015

5.
P. J. CALLAHAN, M. V. HILDEBRANDT, J. R. WEEKS, A census of cusped hyperbolic $3$-manifolds. With microfiche supplement, Math. Comp. 68 (1999), 321-332. MR 99c:57035

6.
D. B. A. EPSTEIN, R. C. PENNER, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988), 67-80. MR 89a:57020

7.
R. FRIGERIO, B. MARTELLI, C. PETRONIO, Small hyperbolic $3$-manifolds with geodesic boundary, math.GT/0211425.

8.
M. FUJII, Hyperbolic $3$-manifolds with totally geodesic boundary which are decomposed into hyperbolic truncated tetrahedra, Tokyo J. Math. 13 (1990), 353-373. MR 92a:57043

9.
S. KOJIMA, Polyhedral decomposition of hyperbolic manifolds with boundary, Proc. Work. Pure Math. 10 (1990), 37-57.

10.
S. KOJIMA, Polyhedral decomposition of hyperbolic $3$-manifolds with totally geodesic boundary, In: ``Aspects of low-dimensional manifolds, Kinokuniya, Tokyo'', Adv. Stud. Pure Math. 20 (1992), 93-112. MR 94c:57023

11.
S. V. MATVEEV, Transformations of special spines, and the Zeeman conjecture, Math. USSR-Izv. 31 (1988), 423-434. MR 89d:57014

12.
R. PIERGALLINI, Standard moves for standard polyhedra and spines, In: ``Third National Conference on Topology, Trieste, 1986'', Rend. Circ. Mat. Palermo (2) Suppl. 18 (1988), 391-414. MR 89k:57003

13.
M. SAKUMA, J.R. WEEKS, The generalized tilt formula, Geom. Dedicata 55 (1995), 115-123. MR 96d:57012

14.
W. P. THURSTON, ``The Geometry and Topology of $\,3$-manifolds'', mimeographed notes, Princeton, 1979.

15.
W. P. THURSTON, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357-381. MR 83h:57019

16.
J.L. TOLLEFSON, Involutions of sufficiently large $3$-manifolds, Topology 20 (1981), 323-352. MR 82h:57014

17.
V. G. TURAEV, O.YA. VIRO, State sum invariants of $3$-manifolds and quantum $6j$-symbols, Topology 31 (1992), 865-902. MR 94d:57044

18.
A. USHIJIMA, A unified viewpoint about geometric objects in hyperbolic space and the generalized tilt formula, In: ``Hyperbolic spaces and related topics, II, Kyoto, 1999'', Surikaisekikenkyusho Kokyuroku

1163 (2000), 85-98.

19.
J. R. WEEKS, Convex hulls and isometries of cusped hyperbolic $3$-manifolds, Topology Appl. 52 (1993), 127-149. MR 95a:57021

20.
J.R. WEEKS, SnapPea, The hyperbolic structures computer program, available from www.northnet.org/weeks.

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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: 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
Posted: August 26, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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