Solvable fundamental groups of compact $3$-manifolds
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- by Benny Evans and Louise Moser
- Trans. Amer. Math. Soc. 168 (1972), 189-210
- DOI: https://doi.org/10.1090/S0002-9947-1972-0301742-6
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Abstract:
A classification is given for groups which can occur as the fundamental group of some compact $3$-manifold. In most cases we are able to determine the topological structure of a compact $3$-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed $3$-manifolds to the more general class of compact $3$-manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a $3$-manifold is a subgroup of the additive group of rationals.References
- D. B. A. Epstein, Projective planes in $3$-manifolds, Proc. London Math. Soc. (3) 11 (1961), 469–484. MR 152997, DOI 10.1112/plms/s3-11.1.469
- R. J. Gregorac, On residually finite generalized free products, Proc. Amer. Math. Soc. 24 (1970), 553–555. MR 260878, DOI 10.1090/S0002-9939-1970-0260878-2
- Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
- Wolfgang Heil, On $P^{2}$-irreducible $3$-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775. MR 251731, DOI 10.1090/S0002-9904-1969-12283-4
- William Jaco, Finitely presented subgroups of three-manifold groups, Invent. Math. 13 (1971), 335–346. MR 300279, DOI 10.1007/BF01406083
- Irving Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. MR 0065561 H. Kneser, Geschlossen Flachen in dreidimensionalem Mannigfaltigheiten, Jber. Deutsch. Math.-Verein. 38 (1929), 248-260. S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. Theorem 7.1, p. 343. MR 28 #122.
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
- William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390
- John Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR 90056, DOI 10.2307/2372566
- Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), no. 1, 189–358 (German). MR 1555256, DOI 10.1007/BF02421324
- H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238 (German). MR 1555366, DOI 10.1007/BF02398271
- John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12–19. MR 121796, DOI 10.2307/1970146 —, On fibering certain $3$-manifolds, Topology of $3$-Manifolds and Related Topics (Proc. The Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95-100. MR 28 #1600.
- Charles Thomas, Nilpotent groups and compact $3$-manifolds, Proc. Cambridge Philos. Soc. 64 (1968), 303–306. MR 233359, DOI 10.1017/s0305004100042857
- Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
- Friedhelm Waldhausen, Gruppen mit Zentrum und $3$-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505–517 (German). MR 236930, DOI 10.1016/0040-9383(67)90008-0
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 168 (1972), 189-210
- MSC: Primary 57A65; Secondary 20E40
- DOI: https://doi.org/10.1090/S0002-9947-1972-0301742-6
- MathSciNet review: 0301742