Compactifying sufficiently regular covering spaces of compact 3-manifolds

Myers, Robert

doi:10.1090/S0002-9939-00-05109-1

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ISSN 1088-6826(online) ISSN 0002-9939(print)

Compactifying sufficiently regular covering spaces of compact 3-manifolds

Author: Robert Myers
Journal: Proc. Amer. Math. Soc. 128 (2000), 1507-1513
MSC (1991): Primary 57M10; Secondary 57N10, 57M60
Posted: February 7, 2000
MathSciNet review: 1637416
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Abstract: In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, $\mathbf{P}^2$ -irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite cyclic and has normalizer a non-finitely-generated subgroup of the rational numbers. In the first case additional information is obtained which is then used to relate Thurston's hyperbolization and virtual bundle conjectures to some algebraic conjectures about certain 3-manifold groups.

1. Mladen Bestvina and Geoffrey Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), no. 3, 469–481. MR 1096169 (93j:20076), http://dx.doi.org/10.1090/S0894-0347-1991-1096169-1
2. Francis Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953 (88c:57013), http://dx.doi.org/10.2307/1971388
3. Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353 (96f:57011), http://dx.doi.org/10.1007/BF01231540
4. D. B. A. Epstein, Projective planes in 3-manifolds, Proc. London Math. Soc. (3) 11 (1961), 469–484. MR 0152997 (27 #2968)
5. Benny Evans and William Jaco, Varieties of groups and three-manifolds, Topology 12 (1973), 83–97. MR 0322846 (48 #1207)
6. David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862 (93m:20065), http://dx.doi.org/10.2307/2946597
7. David Gabai, Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc. 7 (1994), no. 1, 193–198. MR 1205445 (94b:57016), http://dx.doi.org/10.1090/S0894-0347-1994-1205445-3
8. David Gabai and William H. Kazez, 3-manifolds with essential laminations are covered by solid tori, J. London Math. Soc. (2) 47 (1993), no. 3, 557–576. MR 1214916 (94c:57028), http://dx.doi.org/10.1112/jlms/s2-47.3.557
9. D. Gabai, R. Meyerhoff, and N. Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic, MSRI Preprint Series #1996-058.
10. David Gabai and Ulrich Oertel, Essential laminations in 3-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. MR 1005607 (90h:57012), http://dx.doi.org/10.2307/1971476
11. C. McA. Gordon and Wolfgang Heil, Cyclic normal subgroups of fundamental groups of 3-manifolds, Topology 14 (1975), no. 4, 305–309. MR 0400232 (53 #4067)
12. Joel Hass, Hyam Rubinstein, and Peter Scott, Compactifying coverings of closed 3-manifolds, J. Differential Geom. 30 (1989), no. 3, 817–832. MR 1021374 (91d:57009)
13. John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J., 1976. Ann. of Math. Studies, No. 86. MR 0415619 (54 #3702)
14. John Hempel and William Jaco, Fundamental groups of 3-manifolds which are extensions, Ann. of Math. (2) 95 (1972), 86–98. MR 0287550 (44 #4754)
15. Wolfgang Heil, On 𝑃²-irreducible 3-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775. MR 0251731 (40 #4958), http://dx.doi.org/10.1090/S0002-9904-1969-12283-4
16. William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450 (81k:57009)
17. William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484 (84f:53053), http://dx.doi.org/10.2307/2007026
18. G. Mess, Centers of 3-manifold groups and groups which are coarse quasi-isometric to planes, preprint.
19. Michael L. Mihalik, Compactifying coverings of 3-manifolds, Comment. Math. Helv. 71 (1996), no. 3, 362–372. MR 1418943 (97k:57020), http://dx.doi.org/10.1007/BF02566425
20. Michael L. Mihalik and Steven T. Tschantz, Tame combings of groups, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4251–4264. MR 1390045 (97m:20049), http://dx.doi.org/10.1090/S0002-9947-97-01772-8
21. G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J., 1973. Annals of Mathematics Studies, No. 78. MR 0385004 (52 #5874)
22. Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996), x+159 (French, with French summary). MR 1402300 (97e:57013)
23. John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730 (95j:57011)
24. G. P. Scott, Finitely generated 3-manifold groups are finitely presented, J. London Math. Soc. (2) 6 (1973), 437–440. MR 0380763 (52 #1660)
25. G. P. Scott, Compact submanifolds of 3-manifolds, J. London Math. Soc. (2) 7 (1973), 246–250. MR 0326737 (48 #5080)
26. Peter Scott, Normal subgroups in 3-manifold groups, J. London Math. Soc. (2) 13 (1976), no. 1, 5–12. MR 0402751 (53 #6565)
27. Peter Scott, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980), no. 2, 241–277. MR 564473 (81f:57006), http://dx.doi.org/10.2307/2374238
28. Peter B. Shalen, Infinitely divisible elements in 3-manifold groups, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 293–335. Ann. of Math. Studies, No. 84. MR 0375280 (51 #11476)
29. Jonathan Simon, Compactification of covering spaces of compact 3-manifolds, Michigan Math. J. 23 (1976), no. 3, 245–256 (1977). MR 0431176 (55 #4178)
30. John Stallings, On fibering certain 3-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95–100. MR 0158375 (28 #1600)
31. John R. Stallings and S. M. Gersten, Casson’s idea about 3-manifolds whose universal cover is 𝑅³, Internat. J. Algebra Comput. 1 (1991), no. 4, 395–406. MR 1154440 (93b:57018), http://dx.doi.org/10.1142/S0218196791000274
32. 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 (83h:57019), http://dx.doi.org/10.1090/S0273-0979-1982-15003-0
33. Thomas W. Tucker, Non-compact 3-manifolds and the missing-boundary problem, Topology 13 (1974), 267–273. MR 0353317 (50 #5801)
34. Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 0224099 (36 #7146)

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