Limit sets of Kleinian groups and conformally flat Riemannian manifolds
type: Item
instance of: scholarly article
title: Limit sets of Kleinian groups and conformally flat Riemannian manifolds
zbMATH Open document ID: 0854.53035
DOI: 10.1007/BF01231457
author: Hiroyasu Izeki
published in: Inventiones Mathematicae
publication date: 1997-01-22T00:00:00Z
full work available at URL: https://eudml.org/doc/144333
review text: 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 \textit{B. Apanasov} [Int. J. Math. 2, No. 4, 361-382 (1991; Zbl 0761.53010), \textit{R. Kulkarni} and \textit{U. Pinkall} [Math. Z. 216, No. 1, 89-129 (1994; Zbl 0813.53022)], and \textit{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 \textit{R. Schoen} and \textit{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 \textit{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 \textit{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…
reviewed by: Boris N. Apanasov
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 871781
zbMATH Keywords: conformally flat manifolds
zbMATH Keywords: uniformization
zbMATH Keywords: holonomy groups
zbMATH Keywords: Kleinian grous
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: quasifuchsian groups
Wikidata QID: Q115393814
MaRDI profile type: Publication
cites work: Q4692972
cites work: The Exponent of Convergence of Poincaré Series
cites work: The Boundary of Negatively Curved Groups
cites work: Hausdorff dimension of quasi-circles
cites work: A Kaluza-Klein approach to hyperbolic three-manifolds
cites work: Q3708977
cites work: Q3714302
cites work: Group completions and limit sets of Kleinian groups
cites work: A geometric reformulation of 4-dimensional surgery
cites work: Four-manifolds with positive isotropic curvature
cites work: On the decomposition of conformally flat manifolds
cites work: A deformation of flat conformal structures
cites work: The Teichmüller distance on the space of flat conformal structures
cites work: On conformally-flat spaces in the large
cites work: On compact conformally euclidean spaces of dimension \(>2\)
cites work: Q3808798
cites work: On Klein's Combination Theorem
cites work: A characterization of Schottky groups
cites work: Q4039828
cites work: Q3135255
cites work: Relative neighborhoods and the theorems of Smale
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
cites work: Patterson-Sullivan measure and conformally flat metrics
cites work: Q4219000
cites work: Q3993094
cites work: The exponent of convergence of Poincaré series
cites work: The limit set of a Fuchsian group
cites work: Conformally flat manifolds, Kleinian groups and scalar curvature
cites work: Q3856928
cites work: Whitehead torsion of free products
cites work: On torsion-free groups with infinitely many ends
cites work: The density at infinity of a discrete group of hyperbolic motions
cites work: Discrete conformal groups and measurable dynamics
cites work: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
cites work: On isomorphisms of geometrically finite Möbius groups
cites work: A rigidity theorem for Möbius groups
Recommended article: Q4272426
Recommended article: Q4247245
Recommended article: Compact deformations of Fuchsian groups
Recommended article: Limits of quasi-Fuchsian groups with small bending
Recommended article: Quasiconformal stability of Kleinian groups and an embedding of a space of flat conformal structures
Recommended article: Kleinian groups with small limit sets
Recommended article: Infinite group actions on spheres
Recommended article: Deformations of conformal structures on hyperbolic manifolds
Recommended article: An explicit counterexample to the equivariant 𝐾=2 conjecture
Recommended article: The Poincaré metric and a conformal version of a theorem of Thurston
description: scientific article; zbMATH DE number 871781