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Limit sets of Kleinian groups and conformally flat Riemannian manifolds. (English) Zbl 0854.53035
The author studies relations between properties of uniformized conformally flat \(n\)-manifolds and some numerical characteristics of their holonomy groups which are Kleinian subgroups \(G\) of the Möbius group \(M(n)\). Due to B. Apanasov [Int. J. Math. 2, No. 4, 361-382 (1991; Zbl 0761.53010), R. Kulkarni and U. Pinkall [Math. Z. 216, No. 1, 89-129 (1994; Zbl 0813.53022)], and S. Nayatani [Math. Z. (1996, to appear)], such manifolds allow smooth Riemannian metrics.
Using a nice relation between the curvature of such a metric and the critical exponent of the corresponding Kleinian grous \(G \subset M(n)\), which is closely related to the Hausdorff dimension \(\dim_H \Lambda(G)\) of the limit set \(\Lambda(G) \subset S^n\) [see R. Schoen and S. T. Yau, Invent. Math. 92, No. 1, 47-71 (1988; Zbl 0658.53038) and the above paper by S. Nayatani], as well as a vanishing theorem of cohomology for certain conformally flat manifolds (see S. Nayatani above), the author proves the following main result:
Theorem 5.1. Let \(G_0\), \(G \subset M(n)\) be isomorphic torsion free convex cocompact Kleinian groups, with \(\Lambda(G_0)\) a round sphere \(S^k \subset S^n\). Then the Hausdorff dimension \(\dim_H \Lambda (G) \geq k\), and equality holds iff \(\Lambda(G)\) is a round \(k\)-sphere. Moreover, if \(\dim_H \Lambda(G) = k = (n - 1) \geq 2\), the groups \(G\) and \(G_0\) are conjugate by a Möbius transformation. This provides an affirmative answer to P. Tukia’s conjecture on quasifuchsian groups [Invent. Math. 97, No. 2, 405-431 (1989; Zbl 0674.30038)]. In the special case of quasifuchsian groups obtained by bending deformations in dimension \(n = 3\) and \(k = 2\), this rigidity has been proved by D. Sullivan [Bull. Am. Math. Soc., New Ser. 6, 57-73 (1982; Zbl 0489.58027)].
Furthermore, if the cohomological dimension of a convex cocompact group \(G \subset M(n)\) is \(n\), then the author proves that \(\dim_H \Lambda(G) \geq n -1\), and equality holds iff \(G\) is a Fuchsian group, i.e., \(\Lambda(G) = S^{n-1}\). As another application of his methods and the cited works by Schoen-Yau and Nayatani, the author also provides a classification of certain uniformizable conformally flat manifolds which is reduced to the classification of corresponding Kleinian groups. In particular, he proves that such a compact 3-manifold \(M\) allows a compatible Riemannian metric with negative Ricci curvature, unless \(M\) is covered by either \(S^3\), the 3-torus \(T^3\), or a connected sum \(k(S^1 \times S^2)\) of \(k\) copies of \(S^1 \times S^2\) for some \(k\), or \(\mathbb{R} \times H^2 (-1)\).

MSC:
53C20 Global Riemannian geometry, including pinching
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
53A30 Conformal differential geometry (MSC2010)
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References:
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