The virtual Haken conjecture: Experiments and examples

Geometry and Topology

Volumn 7, Issue , 2003, Pages 399-441

  Dunfield, Nathan M ^a   Thurston, William P ^b  

^a HARVARD UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Dehn filling; Experimental evidence; Figure 8 knot; One relator quotients; Virtual Haken Conjecture

Indexed keywords

EID: 3042652298     PISSN: 14653060     EISSN: None     Source Type: Journal    
DOI: 10.2140/gt.2003.7.399     Document Type: Article

References (57)

1. B Baumslag, S J Pride, Groups with two more generators than relators, J. London Math. Soc. 17 (1978) 425-426
2. G Baumslag, J W Morgan, P B Shalen, Generalized triangle groups, Math. Proc. Cambridge Philos. Soc. 102 (1987) 25-31
3. 3, Trans. Amer. Math. Soc. 213 (1975) 315-352
4. W Bosma, J Cannon, Handbook of MAGMA functions, (1994)
5. S Boyer, X Zhang, Virtual Haken 3-manifolds and Dehn filling, Topology 39 (2000) 103-114
6. K S Brown, Buildings, Springer-Verlag, New York (1989)
7. D Calegari, Promoting essential laminations, arXiv:math.GT/0210148
8. P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic 3-manifolds, Math. Comp. 68 (1999) 321-332
9. A Casson and D Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994) 441-456
10. S Choi and W Goldman, The deformation spaces of projectively flat structures on 2-orbifolds, preprint (2001) arXiv:math.GT/0107193
11. J H Conway, R T Curtis, S P Norton, R A Parker, R A Wilson, Atlas of finite groups, Oxford University Press, Eynsham (1985)
12. D Cooper, D D Long, Virtually Haken Dehn-filling, J. Differential Geom. 52 (1999) 173-187
13. D Cooper, D D Long, Some surface subgroups survive surgery, Geometry and Topology 5 (2001) 347-367
14. M Culler, P B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537-545
15. M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237-300
16. J D Dixon, Computing irreducible representations of groups, Math. Comp. 24 (1970) 707-712
17. J D Dixon, Exact solution of linear equations using p-adic expansions, Numer. Math. 40 (1982) 137-141
18. N M Dunfield, Alexander and Thurston norms of 3-manifolds fibering over the circle, Pacific J. Math 200 (2001) 43-58, arXiv:math.GT/9908050
19. N M Dunfield, Which small volume hyperbolic 3-manifolds are Haken? Slides from a talk at the University of Warwick (1999) available from http://www.math.harvard.edu/~nathand
20. N M Dunfield, W P Thurston, Data on the virtual Haken conjecture, http://www.computop.org/software/virtual haken
21. N M Dunfield, W P Thurston, Random 3-manifolds: Heuristics for the Virtual Haken Conjecture, in preparation
22. B Fine, F Roehl, G Rosenberger, A Freiheitssatz for certain one-relator amalgamated products, In: "Combinatorial and geometric group theory (Edinburgh, 1993)", pages 73-86. Cambridge Univ. Press, 1995
23. B Fine, G Rosenberger, Algebraic generalizations of discrete groups, Marcel Dekker Inc. New York (1999)
24. B Freedman, M H Freedman, Kneser-Haken finiteness for bounded 3-manifolds locally free groups, and cyclic covers, Topology 37 (1998) 133-147
25. D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447-510
26. D Gabai, On the geometric and topological rigidity of hyperbolic 3-manifolds, J. Amer. Math. Soc. 10 (1997) 37-74
27. D Gabai, R Meyerho , N Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic, preprint, to appear in Ann. of Math.
28. The GAP Group, Aachen, St Andrews, GAP - Groups, Algorithms, and Programming, Version 4.2, (2000) http://www-gap.dcs.st-and.ac.uk/~gap
29. O Goodman, Snap, http://www.ms.unimelb.edu.au/~snap
30. A Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982) 373-377
31. G Havas, D F Holt, S Rees, Recognizing badly presented Z-modules, Linear Algebra Appl. 192 (1993) 137-163
32. G Havas, B S Majewski, Hermite normal form computation for integer matrices, from: "Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994)", Congr. Numer. 105 (1994) 87-96
33. G Havas, B S Majewski, Integer matrix diagonalization, J. Symbolic Comput. 24 (1997) 399-408
34. M Hildebrand, J Weeks, A computer generated census of cusped hyperbolic 3-manifolds, from: "computers and mathematics (Cambridge, MA, 1989)", Springer (1989) 53-59
35. D F Holt, W Plesken, A cohomological criterion for a finitely presented group to be in finite, J. London Math. Soc. (2) 45 (1992) 469-480
36. J Hoste, M Thistlethwaite, Knotscape, www.math.utk.edu/~morwen
37. W Jaco, U Oertel, An algorithm to decide if a 3-manifold is a Haken manifold, Topology 23 (1984) 195-209
38. R Kirby, Problems in low-dimensional topology, from: "Geometric topology (Athens, GA, 1993)", Amer. Math. Soc. Providence, RI (1997) 35-473, http://www.math.berkeley.edu/~kirby/
39. A Lubotzky, Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem, Ann. of Math. 144 (1996) 441-452
40. A Lubotzky, Free quotients and the first Betti number of some hyperbolic manifolds, Transform. Groups 1 (1996) 71-82
41. W Meeks, L Simon, S-T Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. 116 (1982) 621-659
42. G Mess, The Seifert fibered space conjecture and groups which are coarse quasiisometric to planes, preprint
43. W Plesken, Presentations and representations of groups, from: "Algorithmic algebra and number theory (Heidelberg, 1997)", Springer, Berlin (1999) 423-434
44. W Plesken, B Souvignier, Constructing rational representations of finite groups, Experiment. Math. 5 (1996) 39-47
45. W Plesken, B Souvignier, Constructing representations of finite groups and applications to finitely presented groups, J. Algebra 202 (1998) 690-703
46. n in algebraic number fields, Mat. Sb. (N.S.) 75 (117) (1968) 171-177
47. J D Rogawski, Automorphic representations of unitary groups in three variables, Princeton University Press, Princeton, NJ (1990)
48. n in algebraic number fields, J. Reine Angew. Math. 268/269 (1974) 27-33
49. 1 , Ann. of Math. (2) 117 (1983) 35-70
50. J-P Serre, Trees, Springer-Verlag, Berlin (1980) translated from the French by John Stillwell
51. C T Simpson, Products of matrices, from: "Differential geometry, global analysis, and topology (Halifax, NS, 1990)", Amer. Math. Soc. Providence, RI (1991) 157-185.
52. C C Sims, Computation with finitely presented groups, Cambridge University Press, Cambridge (1994)
53. W P. Thurston, The geometry and topology of 3-manifolds, Lecture notes (1978) http://www.msri.org/publications/books/gt3m/
54. F Waldhausen, The word problem in fundamental groups of sufficiently large irreducible 3-manifolds, Ann. of Math. 88 (1968) 272-280
55. C T C Wall, Rational Euler characteristics, Proc. Cambridge Philos. Soc. 57 (1961) 182-184
56. J Weeks, SnapPea, http://www.geometrygames.org/
57. Y Q Wu, Incompressibility of surfaces in surgered 3-manifolds, Topology 31 (1992) 271-279