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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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