On isomorphisms of geometrically finite Möbius groups
type: Item
instance of: scholarly article
title: On isomorphisms of geometrically finite Möbius groups
zbMATH Open document ID: 0572.30036
DOI: 10.1007/BF02698805
author: Pekka Tukia
published in: Publications Mathématiques
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104003
review text: The goal of the paper is to prove that any type-preserving isomorphism \(j: G\to G'\) of geometrically finite Möbius groups G and G' in \(\bar R^ n=R^ n\cup (\infty)\) (\(G,G'\) are discrete and have finite-sided fundamental polyhedra in the half-space \(R_+^{n+1})\) is induced by a unique homeomorphism \(f_ j: L(G)\to L(G')\) of the limit sets, i.e. \(f_ j(g(x))=j(g)(f_ j(x))\) for \(g\in G\) and \(x\in L(G)\). This result (Th. 3.3) is contained in the section 3. In his proof an important role is played by numerous statements on geometrically finite groups contained in section 2 (it takes nearly half of the paper). Note that these results are greatly covered by the reviewer's earlier papers [Sib. Mat. Zh. 23, No.6, 16-27 (1982; Zbl 0519.30038), Ann. Global Anal. Geom. 1, No.3, 1-22 (1983; Zbl 0531.57012)], see also the reviewer's book ''Discrete transformation groups and manifold structures'', (Russian) (1983; Zbl 0571.57002). Another important result (Th. 3.8) of the paper shows that if \(L(G)\neq \bar R^ n\) and if \(f: \bar R^ n-L(G)\to \bar R^ n-L(G')\) is a homeomorphism inducing j, then j is type-preserving if \(n\geq 2\) and that then f and \(f_ j\) define together a homeomorphism \(f'\) inducing j; for \(n\geq 2\) \(f'\) is quasiconformal if f is and the dilatation is not increased in the extension to \(L(G)\). In particular, for conformal f(and hence f' is a Möbius transformation), this result is consistent with Mostow's rigidity theorem for \(L(G)=\bar R^ n\) (see \textit{G. D. Mostow}, ''Strong rigidity of locally symmetric spaces'' (1973; Zbl 0265.53039), as was already observed by \textit{A. Marden} [Ann. of Math., II. Ser. 99, 383- 462 (1974; Zbl 0282.30014)] for \(n=2\). Note that not only geometrically finite groups possess such rigidity. The exact class of rigidity in the groups of this sense in \(\bar R^ n\), \(n\geq 3\), is indicated by the reviewer [Analytic functions, Proc. Conf., Błażejewko/Pol. 1982, Lect. Notes Math. 1039, 1-8 (1983; Zbl 052…
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 3912724
zbMATH Keywords: type-preserving isomorphism
zbMATH Keywords: geometrically finite Möbius groups
zbMATH Keywords: Mostow's rigidity theorem
reviewed by: Boris N. Apanasov
MaRDI profile type: Publication
cites work: FUNDAMENTAL POLYHEDRONS AND LIMIT POINT SETS OF KLEINIAN GROUPS
cites work: Geometrically finite hyperbolic structures on manifolds
cites work: Limit points of Kleinian groups and finite sided fundamental polyhedra
cites work: Group completions and limit sets of Kleinian groups
cites work: Quasiconformally homogeneous domains
cites work: Q5794159
cites work: Q3871581
cites work: Homotopy equivalences of 3-manifolds with boundaries
cites work: Q5613346
cites work: Q5685358
cites work: The geometry of finitely generated Kleinian groups
cites work: On the isomorphism theorem for Kleinian groups
cites work: On boundaries of Teichmüller spaces and on Kleinian groups. II
cites work: Isomorphisms of function groups
cites work: On the classification of Kleinian groups II-signatures
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: Strong rigidity of Q-rank 1 lattices
cites work: Invariant extension of quasiconformal deformations
cites work: Q5631380
cites work: Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group
cites work: The Hausdorff dimension of the limit set of a geometrically finite Kleinian group
cites work: Differentiability and rigidity of Möbius groups
cites work: Quasisymmetric embeddings of metric spaces
cites work: Lipschitz and quasiconformal approximation and extension
cites work: Quasiconformal extension from dimension n to n+1
cites work: Discrete Moebius Groups: Fundamental Polyhedra and Convergence
cites work: The Cohomology of Homogeneous Spaces
description: scientific article; zbMATH DE number 3912724