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

Full-text PDF Free Access

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.

**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 -manifolds*,`math.GT/0101234`.**3.**Alan F. Beardon,*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777****4.**Riccardo Benedetti and Carlo Petronio,*Lectures on hyperbolic geometry*, Universitext, Springer-Verlag, Berlin, 1992. MR**1219310****5.**Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks,*A census of cusped hyperbolic 3-manifolds*, Math. Comp.**68**(1999), no. 225, 321–332. With microfiche supplement. MR**1620219**, 10.1090/S0025-5718-99-01036-4**6.**D. B. A. Epstein and R. C. Penner,*Euclidean decompositions of noncompact hyperbolic manifolds*, J. Differential Geom.**27**(1988), no. 1, 67–80. MR**918457****7.**R. FRIGERIO, B. MARTELLI, C. PETRONIO,*Small hyperbolic -manifolds with geodesic boundary*,`math.GT/0211425`.**8.**Michihiko Fujii,*Hyperbolic 3-manifolds with totally geodesic boundary which are decomposed into hyperbolic truncated tetrahedra*, Tokyo J. Math.**13**(1990), no. 2, 353–373. MR**1088237**, 10.3836/tjm/1270132267**9.**S. KOJIMA,*Polyhedral decomposition of hyperbolic manifolds with boundary*, Proc. Work. Pure Math.**10**(1990), 37-57.**10.**Sadayoshi Kojima,*Polyhedral decomposition of hyperbolic 3-manifolds with totally geodesic boundary*, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math., vol. 20, Kinokuniya, Tokyo, 1992, pp. 93–112. MR**1208308****11.**S. V. Matveev,*Transformations of special spines, and the Zeeman conjecture*, Izv. Akad. Nauk SSSR Ser. Mat.**51**(1987), no. 5, 1104–1116, 1119 (Russian); English transl., Math. USSR-Izv.**31**(1988), no. 2, 423–434. MR**925096****12.**Riccardo Piergallini,*Standard moves for standard polyhedra and spines*, Rend. Circ. Mat. Palermo (2) Suppl.**18**(1988), 391–414. Third National Conference on Topology (Italian) (Trieste, 1986). MR**958750****13.**Makoto Sakuma and Jeffrey R. Weeks,*The generalized tilt formula*, Geom. Dedicata**55**(1995), no. 2, 115–123. MR**1334208**, 10.1007/BF01264924**14.**W. P. THURSTON, ``The Geometry and Topology of -manifolds'', mimeographed notes, Princeton, 1979.**15.**William P. Thurston,*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 3, 357–381. MR**648524**, 10.1090/S0273-0979-1982-15003-0**16.**Jeffery L. Tollefson,*Involutions of sufficiently large 3-manifolds*, Topology**20**(1981), no. 4, 323–352. MR**617370**, 10.1016/0040-9383(81)90018-5**17.**V. G. Turaev and O. Ya. Viro,*State sum invariants of 3-manifolds and quantum 6𝑗-symbols*, Topology**31**(1992), no. 4, 865–902. MR**1191386**, 10.1016/0040-9383(92)90015-A**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.**Jeffrey R. Weeks,*Convex hulls and isometries of cusped hyperbolic 3-manifolds*, Topology Appl.**52**(1993), no. 2, 127–149. MR**1241189**, 10.1016/0166-8641(93)90032-9**20.**J.R. WEEKS,*SnapPea*, The hyperbolic structures computer program, available from`www.northnet.org/weeks`.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57M50,
57M25

Retrieve articles in all journals with MSC (2000): 57M50, 57M25

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

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