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

DOI:
https://doi.org/10.1090/S0002-9939-00-05109-1

Published electronically:
February 7, 2000

MathSciNet review:
1637416

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, -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.**M. Bestvina and G. Mess,*The boundary of negatively curved groups*, J. Amer. Math. Soc. 4 (1991), no. 3, 469-481. MR**93j:20076****2.**F. Bonahon,*Bouts des variétés hyperboliques de dimension*Ann. of Math. (2) 124 (1986), no. 1, 71-158. MR**88c:57013****3.**A. Casson and D. Jungreis,*Convergence groups and Seifert fibered 3-manifolds*, Invent. Math. 118 (1994), 441-456. MR**96f:57011****4.**D. B. A. Epstein,*Projective planes in**-manifolds*, Proc. London Math. Soc. (3) 11 1961 469-484. MR**27:2968****5.**B. Evans and W. Jaco,*Varieties of groups and three-manifolds*, Topology 12 (1973), 83-97. MR**48:1207****6.**D. Gabai,*Convergence groups are Fuchsian groups*, Annals of Math. 136 (1992), 447-510. MR**93m:20065****7.**D. Gabai,*Homotopy hyperbolic 3-manifolds are virtually hyperbolic*, J. Amer. Math. Soc. 7 (1994), 193-198. MR**94b:57016****8.**D. Gabai and W. Kazez,*-manifolds with essential laminations are covered by solid tori*, J. London Math. Soc. (2) 47 (1993), no. 3, 557-576. MR**94c:57028****9.**D. Gabai, R. Meyerhoff, and N. Thurston,*Homotopy hyperbolic 3-manifolds are hyperbolic*, MSRI Preprint Series #1996-058.**10.**D. Gabai and U. Oertel,*Essential laminations in 3-manifolds*, Ann. Math. 130 (1989), 41-73. MR**90h:57012****11.**C. Gordon and W. Heil,*Cyclic normal subgroups of fundamental groups of**-manifolds*, Topology 14 (1975), no. 4, 305-309. MR**53:4067****12.**J. Hass, H. Rubinstein, and P. Scott,*Compactifying coverings of closed 3-manifolds*, J. Differential Geometry 30 (1989), 817-832. MR**91d:57009****13.**J. Hempel,*3-Manifolds*, Ann. of Math. Studies, No. 86, Princeton, (1976). MR**54:3702****14.**J. Hempel and W. Jaco,*Fundamental groups of**-manifolds which are extensions*, Ann. of Math. (2) 95 (1972), 86-98. MR**44:4754****15.**W. Heil,*On**-irreducible**-manifolds*, Bull. Amer. Math. Soc. 75, (1969), 772-775. MR**40:4958****16.**W. Jaco,*Lectures on three-manifold topology*, CBMS Regional Conference Series in Math., No. 43, Amer. Math. Soc. (1980). MR**81k:57009****17.**W. Meeks, L. Simon, S. T. Yau,*Embedded minimal surfaces, exotic spheres, and manifolds with postive Ricci curvature*, Annals of Math, 116 (1982), 621-659. MR**84f:53053****18.**G. Mess,*Centers of 3-manifold groups and groups which are coarse quasi-isometric to planes*, preprint.**19.**M. Mihalik,*Compactifying coverings of**-manifolds*, Comment. Math. Helv. 71 (1996), no. 3, 362-372. MR**97k:57020****20.**M. Mihalik and S. Tschantz,*Tame combings of groups*, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4251-4264. MR**97m:20049****21.**G. D. Mostow,*Strong rigidity of locally symmetric spaces*, Annals of Mathematics Studies, No. 78. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR**52:5874****22.**Otal, J-P,*Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3,*Astérisque No. 235 (1996). MR**97e:57013****23.**J. Ratcliffe,*Foundations of hyperbolic manifolds*, Graduate Texts in Mathematics, 149. Springer-Verlag, New York, 1994. MR**95j:57011****24.**G. P. Scott,*Finitely generated**-manifold groups are finitely presented*, J. London Math. Soc. (2) 6 (1973), 437-440. MR**52:1660****25.**G. P. Scott,*Compact submanifolds of 3-manifolds*, J. London Math. Soc. 7 (1973), 246-250. MR**48:5080****26.**P. Scott,*Normal subgroups in**-manifold groups*, J. London Math. Soc. (2) 13 (1976), no. 1, 5-12. MR**53:6565****27.**P. Scott,*A new proof of the annulus and torus theorems*, Amer. J. Math. 102 (1980), no. 2, 241-277. MR**81f:57006****28.**P. Shalen,*Infinitely divisible elements in**-manifold groups*, Knots, groups, and -manifolds (Papers dedicated to the memory of R. H. Fox), pp. 293-335. Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975. MR**51:11476****29.**J. Simon,*Compactifications of covering spaces of compact 3-manifolds*, Michigan Math. J. 23 (1976), 245-256. MR**55:4178****30.**J. Stallings,*On fibering certain**-manifolds*, 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 95-100 Prentice-Hall, Englewood Cliffs, N.J. MR**28:1600****31.**J. Stallings and S. Gersten,*Casson's idea about**-manifolds whose universal cover is*Internat. J. Algebra Comput. 1 (1991), no. 4, 395-406. MR**93b:57018****32.**W. P. Thurston,*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR**83h:57019****33.**T. Tucker,*Non-compact 3-manifolds and the missing-boundary problem*, Topology 13 (1974), 267-273. MR**50:5801****34.**F. Waldhausen,*On irreducible 3-manifolds which are sufficiently large*, Ann. of Math., 87 (1968), 56-88. MR**36:7146**

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Additional Information

**Robert Myers**

Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
myersr@math.okstate.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05109-1

Keywords:
3-manifold,
covering space,
compactification,
hyperbolic 3-manifold

Received by editor(s):
October 7, 1997

Received by editor(s) in revised form:
June 1, 1998

Published electronically:
February 7, 2000

Additional Notes:
Research at MSRI is supported in part by NSF grant DMS-9022140.

Communicated by:
Ronald A. Fintushel

Article copyright:
© Copyright 2000
American Mathematical Society