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

Published electronically:
February 7, 2000

MathSciNet review:
1637416

Full-text PDF Free Access

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.**Mladen Bestvina and Geoffrey Mess,*The boundary of negatively curved groups*, J. Amer. Math. Soc.**4**(1991), no. 3, 469–481. MR**1096169**, 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**, 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**, 10.1007/BF01231540**4.**D. B. A. Epstein,*Projective planes in 3-manifolds*, Proc. London Math. Soc. (3)**11**(1961), 469–484. MR**0152997****5.**Benny Evans and William Jaco,*Varieties of groups and three-manifolds*, Topology**12**(1973), 83–97. MR**0322846****6.**David Gabai,*Convergence groups are Fuchsian groups*, Ann. of Math. (2)**136**(1992), no. 3, 447–510. MR**1189862**, 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**, 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**, 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**, 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****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****13.**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****14.**John Hempel and William Jaco,*Fundamental groups of 3-manifolds which are extensions*, Ann. of Math. (2)**95**(1972), 86–98. MR**0287550****15.**Wolfgang Heil,*On 𝑃²-irreducible 3-manifolds*, Bull. Amer. Math. Soc.**75**(1969), 772–775. MR**0251731**, 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****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**, 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**, 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**, 10.1090/S0002-9947-97-01772-8**21.**G. D. Mostow,*Strong rigidity of locally symmetric spaces*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. MR**0385004****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****23.**John G. Ratcliffe,*Foundations of hyperbolic manifolds*, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR**1299730****24.**G. P. Scott,*Finitely generated 3-manifold groups are finitely presented*, J. London Math. Soc. (2)**6**(1973), 437–440. MR**0380763****25.**G. P. Scott,*Compact submanifolds of 3-manifolds*, J. London Math. Soc. (2)**7**(1973), 246–250. MR**0326737****26.**Peter Scott,*Normal subgroups in 3-manifold groups*, J. London Math. Soc. (2)**13**(1976), no. 1, 5–12. MR**0402751****27.**Peter Scott,*A new proof of the annulus and torus theorems*, Amer. J. Math.**102**(1980), no. 2, 241–277. MR**564473**, 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****29.**Jonathan Simon,*Compactification of covering spaces of compact 3-manifolds*, Michigan Math. J.**23**(1976), no. 3, 245–256 (1977). MR**0431176****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****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**, 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**, 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****34.**Friedhelm Waldhausen,*On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**0224099**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
57M10,
57N10,
57M60

Retrieve articles in all journals with MSC (1991): 57M10, 57N10, 57M60

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