Algorithmic detection and description of hyperbolic structures on closed 3–manifolds with solvable word problem

Manning, Jason Fox¹

¹ Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106, USA

Geometry & topology, Tome 6 (2002) no. 1, pp. 1-26.

Résumé

We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable $3$-manifold $M$ is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in ${\pi }_1\left(M\right)$.

Détail

DOI : 10.2140/gt.2002.6.1

Keywords: 3–manifold, Kleinian group, word problem, recognition problem, geometric structure

Bibliographie

[1] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics 61, Birkhäuser (1985)

[2] A F Beardon, The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer (1983)

[3] T Becker, V Weispfenning, Gröbner bases, Graduate Texts in Mathematics 141, Springer (1993)

[4] R H Bing, Locally tame sets are tame, Ann. of Math. $(2)$ 59 (1954) 145

[5] D Cooper, C D Hodgson, S P Kerckhoff, Thurston's Orbifold Theorem, to appear

[6] M Culler, P B Shalen, Varieties of group representations and splittings of 3–manifolds, Ann. of Math. $(2)$ 117 (1983) 109

[7] D B A Epstein, J W Cannon, D F Holt, S V F Levy, M S Paterson, W P Thurston, Word processing in groups, Jones and Bartlett Publishers (1992)

[8] D Gabai, G R Meyerhoff, N Thurston, Homotopy hyperbolic 3–manifolds are hyperbolic, Ann. of Math. $(2)$ 157 (2003) 335

[9] G Hemion, On the classification of homeomorphisms of 2–manifolds and the classification of 3–manifolds, Acta Math. 142 (1979) 123

[10] W Jaco, J L Tollefson, Algorithms for the complete decomposition of a closed 3–manifold, Illinois J. Math. 39 (1995) 358

[11] M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243

[12] R Loos, Computing in algebraic extensions, from: "Computer algebra", Springer (1983) 173

[13] B Maskit, On Poincaré's theorem for fundamental polygons, Advances in Math. 7 (1971) 219

[14] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1998)

[15] E E Moise, Affine structures in 3–manifolds V: The triangulation theorem and Hauptvermutung, Ann. of Math. $(2)$ 56 (1952) 96

[16] G A Niblo, L D Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621

[17] C Petronio, J R Weeks, Partially flat ideal triangulations of cusped hyperbolic 3–manifolds, Osaka J. Math. 37 (2000) 453

[18] R Riley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, Math. Comp. 40 (1983) 607

[19] J H Rubinstein, An algorithm to recognize the 3–sphere, from: "Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)", Birkhäuser (1995) 601

[20] P Scott, The geometries of 3–manifolds, Bull. London Math. Soc. 15 (1983) 401

[21] H Seifert, Komplexe mit Seitenzuordnung, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1975) 49

[22] Z Sela, Topics in 3–manifolds, PhD thesis, Hebrew University (1991)

[23] Z Sela, The isomorphism problem for hyperbolic groups I, Ann. of Math. $(2)$ 141 (1995) 217

[24] A Skinner, The word problem in a class of non–Haken 3–manifolds, Topology 33 (1994) 215

[25] M Stocking, Almost normal surfaces in 3–manifolds, PhD thesis, UC Davis (1996)

[26] A Thompson, Thin position and the recognition problem for $S^3$, Math. Res. Lett. 1 (1994) 613

[27] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. $($N.S.$)$ 6 (1982) 357

[28] F Waldhausen, The word problem in fundamental groups of sufficiently large irreducible 3–manifolds, Ann. of Math. $(2)$ 88 (1968) 272

[29] F Waldhausen, Recent results on sufficiently large 3–manifolds, from: "Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2", Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc. (1978) 21