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Hyperbolic 4-manifolds and conformally flat 3-manifolds. (English) Zbl 0692.57012
The authors discover complete hyperbolic manifold structures on non- trivial plane bundles \(E\to \Sigma\) over an oriented closed surface \(\Sigma\) and uniformized conformal structures (that is, conformally flat structures) on the corresponding non-trivial circle bundles over \(\Sigma\). All these bundles satisfy the inequality \(| \chi (E)| \leq | \chi (\Sigma)|\), and the authors conjecture that it is a necessary condition for the existence of such structures. The authors’ method is based on some construction of discrete groups \(G\subset SO(4,1)\) which have as their limit sets embedded circles \(\Lambda (G)\subset S^ 3\). For the first time, related constructions of “wild” limit curves were contained in the reviewer’s paper “Kleinian groups, Teichmüller space and Mostow’s rigidity theorem” [Sib. Mat. Zh. 21, No.4, 3-15 (1980; Zbl 0499.30032)] and in B. Maskit’s book “Kleinian groups” (1988; Zbl 0627.30039). For their generalizations see Ch. 9, § 5 of the reviewer’s book “The geometry of discrete groups in space and uniformization problems” (Kluwer Acad. Publ. (Math. and Appl. 40), Dordrecht, 1990) and N. Kuiper’s paper reviewed below (see Zbl 0692.57013).
Reviewer: B.N.Apanasov

MSC:
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
22E40 Discrete subgroups of Lie groups
51M10 Hyperbolic and elliptic geometries (general) and generalizations
57S30 Discontinuous groups of transformations
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References:
[1] B. N. Apanasov, Kleinian groups, Teichmüller space and Mostow’s Rigidity Theorem,Sibirsk. Mat. Zh.,21 (1980), no 4, 3–15 (Siberian Math. J.,21 (1980), 483–491). · Zbl 0499.30032
[2] —-,Diskretnye grouppy preobrazovanil i struktury mnogoobrazil (Discrete groups of transformations and manifold structures), Akad. Nauk SSSR, Siberian Section, Novosibirsk, 1983.
[3] W. M. Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3-manifolds,Trans. A.M.S.,278 (1983), 573–583. · Zbl 0518.53041
[4] —-, Projective structures with Fuchsian holonomy,J. Diff. Geom.,25 (1987), 297–326. · Zbl 0595.57012
[5] Y. Kamishima, Conformally flat manifolds whose developing maps are not surjective, I,Trans. A.M.S.,294 (1986), 607–623. · Zbl 0608.53036 · doi:10.1090/S0002-9947-1986-0825725-2
[6] –, Conformally flat manifolds whose developing maps are not surjective, II, to appear.
[7] M. Kapovich,Flat conformal structures on 3-manifolds, Preprint N17, Novosibirsk, 1987. · Zbl 0631.53021
[8] N. H. Kuiper, Hyperbolic manifolds and tesselations,Publ. Math. I.H.E.S.,68 (1988), 47–76. · Zbl 0692.57013
[9] R. Kulkarni, The principle of uniformization,J. Diff. Geom.,13 (1978), 109–138. · Zbl 0381.53023
[10] R. Kulkarni andU. Pinkall, Uniformization of geometric structures with applications to conformal geometry, inDiff. Geo. Peñiscola, Springer Lecture Notes in Math.,1209 (1986), 190–209.
[11] B. Maskit,Kleinian Groups, Grundlehrer der math Wiss., no 287, Springer-Verlag, 1987.
[12] W. S. Massey, Proof of a conjecture of Whitney,Pacific J. Math.,31 (1969), 143–156. · Zbl 0198.56701
[13] W. Thurston,Foliations of 3-manifolds which are circle bundles, Thesis, Univ. of Calif., Berkeley, Calif., 1972.
[14] J. Wood, Bundles with totally disconnected structure group,Comment. Math. Helv.,46 (1971), 257–273. · Zbl 0217.49202 · doi:10.1007/BF02566843
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