Geodesic
Hyperbolic 3-Manifolds with Geodesic Boundary: Enumeration and Volume Calculation
Informatics and Automation, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 167-183.

Voir la notice de l'article provenant de la source Math-Net.Ru

We describe a natural strategy to enumerate compact hyperbolic 3-manifolds with geodesic boundary in increasing order of complexity. We show that the same strategy can be applied in order to analyze simultaneously compact manifolds and finite-volume manifolds with toric cusps. In contrast, we show that if one allows annular cusps, the number of manifolds grows very rapidly and our strategy cannot be employed to obtain a complete list. We also carefully describe how to compute the volume of our manifolds, discussing formulas for the volume of a tetrahedron with generic dihedral angles in a hyperbolic space. 
@article{TRSPY_2006_252_a14,
     author = {A. D. Mednykh and C. Petronio},
     title = {Hyperbolic {3-Manifolds} with {Geodesic} {Boundary:} {Enumeration} and {Volume} {Calculation}},
     journal = {Informatics and Automation},
     pages = {167--183},
     publisher = {mathdoc},
     volume = {252},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_252_a14/}
}
TY  - JOUR
AU  - A. D. Mednykh
AU  - C. Petronio
TI  - Hyperbolic 3-Manifolds with Geodesic Boundary: Enumeration and Volume Calculation
JO  - Informatics and Automation
PY  - 2006
SP  - 167
EP  - 183
VL  - 252
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2006_252_a14/
LA  - en
ID  - TRSPY_2006_252_a14
ER  - 
%0 Journal Article
%A A. D. Mednykh
%A C. Petronio
%T Hyperbolic 3-Manifolds with Geodesic Boundary: Enumeration and Volume Calculation
%J Informatics and Automation
%D 2006
%P 167-183
%V 252
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2006_252_a14/
%G en
%F TRSPY_2006_252_a14
A. D. Mednykh; C. Petronio. Hyperbolic 3-Manifolds with Geodesic Boundary: Enumeration and Volume Calculation. Informatics and Automation, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 167-183. http://geodesic.mathdoc.fr/item/TRSPY_2006_252_a14/

[1] Agol I., “Bounds on exceptional Dehn filling”, Geom. Topol., 4 (2000), 431–449 | DOI | MR | Zbl

[2] Benedetti R., Petronio C., Lectures on hyperbolic geometry, Universitext, Springer, Berlin, 1992 | MR | Zbl

[3] Benedetti R., Petronio C., “A finite graphic calculus for 3-manifolds”, Manuscr. math., 88 (1995), 291–310 | DOI | MR | Zbl

[4] Callahan P.J., Hildebrand M.V., Weeks J.R., “A census of cusped hyperbolic 3-manifolds. With microfiche supplement”, Math. Comput., 68 (1999), 321–332 | DOI | MR | Zbl

[5] Casler B.G., “An imbedding theorem for connected 3-manifolds with boundary”, Proc. Amer. Math. Soc., 16 (1965), 559–566 | DOI | MR | Zbl

[6] Cho Yu., Kim H., “On the volume formula for hyperbolic tetrahedra”, Discr. Comput. Geom., 22 (1999), 347–366 | DOI | MR | Zbl

[7] Connelly R., Sabitov I., Walz A., “The Bellows conjecture”, Contrib. Alg. and Geom., 38 (1997), 1–10 | MR | Zbl

[8] Costantino F., Frigerio R., Martelli B., Petronio C., Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling, arXiv.org: math.GT/0402339 | MR

[9] Derevnin D.A., Mednykh A.D., On the volume of spherical Lambert cube, arXiv.org: math.MG/0212301

[10] Derevnin D.A., Mednykh A.D., “O formule ob'ema giperbolicheskogo tetraedra”, UMN, 60:2 (2005), 159–160 | MR | Zbl

[11] Derevnin D.A., Mednykh A.D., Pashkevich M.G., “Ob'em simmetrichnogo tetraedra v giperbolicheskom i sfericheskom prostranstvakh”, Sib. mat. zhurn., 45:5 (2004), 1022–1031 | MR | Zbl

[12] Epstein D.B.A., Penner R.C., “Euclidean decompositions of noncompact hyperbolic manifolds”, J. Diff. Geom., 27 (1988), 67–80 | MR | Zbl

[13] Fomenko A., Matveev S.V., Algorithmic and computer methods for three-manifolds, Math. and Appl., 425, Kluwer, Dordrecht, 1997 | MR | Zbl

[14] Frigerio R., Martelli B., Petronio C., “Complexity and Heegaard genus of an infinite class of hyperbolic 3-manifolds”, Pacif. J. Math., 210 (2003), 283–297 | DOI | MR | Zbl

[15] Frigerio R., Martelli B., Petronio C., “Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary”, J. Diff. Geom., 64 (2003), 425–455 | MR | Zbl

[16] Frigerio R., Martelli B., Petronio C., “Small hyperbolic 3-manifolds with geodesic boundary”, Exp. Math., 13 (2004), 171–184 | MR | Zbl

[17] Frigerio R., Petronio C., “Construction and recognition of hyperbolic 3-manifolds with geodesic boundary”, Trans. Amer. Math. Soc., 356 (2004), 3243–3282 | DOI | MR | Zbl

[18] Fujii M., “Hyperbolic 3-manifolds with totally geodesic boundary which are decomposed into hyperbolic truncated tetrahedra”, Tokyo J. Math., 13 (1990), 353–373 | DOI | MR | Zbl

[19] Gordon C.McA., “Small surfaces and Dehn filling”, Proc. Kirbyfest (Berkeley (CA), 1998), Geom. and Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999, 177–199 | MR | Zbl

[20] Gordon C.McA., Wu Y.Q., “Annular Dehn fillings”, Comment. Math. Helv., 75 (2000), 430–456 | DOI | MR | Zbl

[21] Gordon C.McA., Wu Y.Q., “Annular and boundary reducing Dehn fillings”, Topology, 39 (2000), 531–548 | DOI | MR | Zbl

[22] Wu-Yi Hsiang, “On infinitesimal symmetrization and volume formula for spherical or hyperbolic tetrahedrons”, Quart. J. Math. Oxford. Ser. 2, 39 (1988), 463–468 | DOI | MR

[23] Johannson K., Homotopy equivalences of 3-manifolds with boundaries, Lect. Notes Math., 761, Springer, Berlin, 1979 | MR | Zbl

[24] Kellerhals R., “On the volume of hyperbolic polyhedra”, Math. Ann., 285 (1989), 541–569 | DOI | MR | Zbl

[25] Kojima S., “Isometry transformations of hyperbolic 3-manifolds”, Topol. and Appl., 29 (1988), 297–307 | DOI | MR | Zbl

[26] Kojima S., “Polyhedral decomposition of hyperbolic manifolds with boundary”, Proc. Work. Pure Math., 10 (1990), 37–57

[27] Kojima S., “Polyhedral decomposition of hyperbolic 3-manifolds with totally geodesic boundary”, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math., 20, Kinokuniya, Tokyo, 1992, 93–112 | MR | Zbl

[28] Lackenby M., “Word hyperbolic Dehn surgery”, Invent. math., 140 (2000), 243–282 | DOI | MR | Zbl

[29] Leibon G., Doyle P., 23040 symmetries of hyperbolic tetrahedra, arXiv.org: math.GT/0309187 | MR

[30] Lobachevskii N.I., “Primeneniya voobrazhaemoi geometrii k nekotorym integralam”, Poln. sobr. soch., t. 3, Izd-vo AN SSSR, M.; L., 1949, 181–294

[31] Martelli B., Complexity of 3-manifolds, arXiv.org: math.GT/0405250 | MR

[32] Martelli B., Petronio C., “3-manifolds up to complexity 9”, Exp. Math., 10 (2001), 207–236 | MR | Zbl

[33] Martin G.J., “The volume of regular tetrahedra and sphere packings in hyperbolic 3-space”, Math. Chron., 20 (1991), 127–147 | MR | Zbl

[34] Matveev S.V., “Complexity theory of three-dimensional manifolds”, Acta appl. math., 19 (1990), 101–130 | MR | Zbl

[35] Matveev S.V., Algorithmic topology and classification of 3-manifolds, Algorithms and Comput. Math., 9, Springer, Berlin, 2003 | MR | Zbl

[36] Mednykh A.D., Parker J., Vesnin A.Yu., On hyperbolic polyhedra arising as convex cores of quasi-Fuchsian punctured torus groups, RIM-GARC Preprint Ser. N 02-01, Seoul Nat. Univ., Seoul, 2002, 33 pp. | MR

[37] Milnor J., “Hyperbolic geometry: the first 150 years”, Bull. Amer. Math. Soc., 6 (1982), 9–24 | DOI | MR | Zbl

[38] Mohanty Y., “The Regge symmetry is a scissors congruence in hyperbolic space”, Alg. and Geom. Topol., 3 (2003), 1–31 | DOI | MR | Zbl

[39] Murakami J., Yano M., On the volume of a hyperbolic and spherical tetrahedron, , 2001 http://www.f.waseda.jp/murakami/papers/tetrahedronrev3.pdf

[40] Paoluzzi L., Zimmermann B., “On a class of hyperbolic 3-manifolds and groups with one defining relation”, Geom. Dedicata, 60 (1996), 113–123 | DOI | MR | Zbl

[41] Petronio C., “Ideal triangulations of hyperbolic 3-manifolds”, Boll. Un. Mat. Ital. Ser. 8. Sez. B: Artic. Ric. Mat., 3 (2000), 657–672 | MR | Zbl

[42] Rourke C., Sanderson B., Introduction to piecewise-linear topology, Ergebn. Math., 69, Springer, New York; Heidelberg, 1972 | MR | Zbl

[43] Sabitov I.Kh., “Ob'em mnogogrannika kak funktsiya dlin ego reber”, Fund. i prikl. matematika, 2:1 (1996), 305–307 | MR | Zbl

[44] Sakuma M., Weeks J.R., “The generalized tilt formula”, Geom. Dedicata, 55 (1995), 115–123 | DOI | MR | Zbl

[45] Scharlemann M., “Heegaard splittings of 3-manifolds”, Low Dimensional Topology, New Stud. Adv. Math., 3, Intern. Press, Somerville (MA), 2003, 25–39 | MR | Zbl

[46] Schläfli L., Theorie der vielfachen Kontinuität, Gesamm. math. Abhandl., Birkhäuser, Basel, 1950

[47] Thurston W.P., The geometry and topology of 3-manifolds, Mimeograph. notes, Princeton, 1979

[48] Turaev V.G., Viro O., “State sum invariants of 3-manifolds and quantum $6j$-symbols”, Topology, 31 (1992), 865–902 | DOI | MR | Zbl

[49] Ushijima A., “A unified viewpoint about geometric objects in hyperbolic space and the generalized tilt formula”, Hyperbolic spaces and related topics. II (Kyoto, 1999), Sūrikaisekikenkyūsho Kōkyūroku, 1163, Kyoto Univ., Kyoto, 2000, 85–98 | MR

[50] Ushijima A., “The tilt formula for generalized simplices in hyperbolic space”, Discr. Comput. Geom., 28 (2002), 19–27 | MR | Zbl

[51] Ushijima A., A volume formula for generalized hyperbolic tetrahedra, arXiv.org: math.GT/0309216

[52] Vinberg E.B., Geometry. II: Spaces of constant curvature, Springer, New York, 1993 | MR | Zbl

[53] Weeks J.R., “Convex hulls and isometries of cusped hyperbolic 3-manifolds”, Topol. and Appl., 52 (1993), 127–149 | DOI | MR | Zbl

[54] Weeks J.R., SnapPea: The hyperbolic structures computer program, http://www.geometrygames.org

[55] Wu Y.Q., “Incompressibility of surfaces in surgered 3-manifolds”, Topology, 31 (1992), 271–279 | DOI | MR | Zbl

[56] Wu Y.Q., “Sutured manifold hierarchies, essential laminations, and Dehn surgery”, J. Diff. Geom., 48 (1998), 407–437 | MR | Zbl

[57] Petronio C., Home page, http://www.dm.unipi.it/pages/petronio/public_html/