Action of Möbius transformations on homeomorphisms: Stability and rigidity
type: Item
instance of: scholarly article
title: Action of Möbius transformations on homeomorphisms: Stability and rigidity
zbMATH Open document ID: 0865.30030
DOI: 10.1007/BF02246768
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-07-17T00:00:00Z
full work available at URL: https://eudml.org/doc/58219
review text: The author studies the action of the group of Möbius transformations on the universal Teichmüller space (defined as the set of all normalized homeomorphisms of the circle at infinity of the hyperbolic plane with the appropriate topology) and its analogues in higher-dimensional hyperbolic spaces (here e.g. instead of circle, he considers spheres at infinity). He extends the characterization of quasisymmetric homeomorphisms to a characterization of quasiconformal homeomorphisms of the sphere at infinity. A homeomorphism \(f:S^1\to S^1\) is called normalized if it fixes three points \(a,b,c\in S^1\), where \(S^1\) is the unit circle of the complex plane \({\mathbf C}\). \(\text{id}:S^1\to S^1\) is considered the distinguished point of the set \(N\) of all normalized homeomorphisms and will denoted by \({\mathbf O}\). Let \(H^n_F\) be one of the hyperbolic spaces over the fields \({\mathbf F}={\mathbf R}\), \({\mathbf C}\), \({\mathbf H}\) of real numbers, complex numbers, quaternions or over the algebra \({\mathbf F}={\mathbf O}\) of Cayley numbers (in this case \(n=2\)). In particular, \(H^n_R=H^n\). Let \(S\) be the sphere at infinity of \(H^n_F\) and let \(H\) be the group of all homeomorphisms \(S\to S\). The group of isometries of \(H^n_f\) acts on \(S\) by homeomorphisms usually called Möbius transformations. This group is denoted by Möb. The first three sections of this paper are restricted to the case \(H^2\). Let \(H\) be the group of orientation-preserving homeomorphisms \(f:S^1\to S^1\) and Möb its subgroup of Möbius transformations. In section 1, he considers the points \(f\in H\) which are stable (i.e. \(f\neq 0\) and its Möb-orbit is closed) or semi-stable (i.e. the closure of its Möb-orbit does not contain \({\mathbf O}\)), while in Section 2, the points of \(H\) fixed by some Fuchsian group, which is equivalent with considering homeomorphisms conjugating one Fuchsian group to another one. In section 3, he establishes the following pointwise non-rigidity…
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 881246
zbMATH Keywords: semi-stable points
zbMATH Keywords: Möbius transformations
zbMATH Keywords: Fuchsian group
author: Nikolai V. Ivanov
reviewed by: Petru Caraman
MaRDI profile type: Publication
cites work: A geometric proof of Mostow's rigidity theorem for groups of divergence type
cites work: Q3805010
cites work: Q3491883
cites work: Conformally natural extension of homeomorphisms of the circle
cites work: Harmonic Analysis in Phase Space. (AM-122)
cites work: Q4026344
cites work: The Nielsen realization problem
cites work: Q3744437
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: Universal constructions in Teichmüller theory
cites work: Strong rigidity of Q-rank 1 lattices
cites work: Q3958740
cites work: Q5607484
cites work: Differentiability and rigidity of Möbius groups
cites work: MOSTOW-RIGIDITY AND NON-COMPACT HYPERBOLIC MANIFOLDS
cites work: Q3114387
OpenAlex ID: W1971721305
description: scientific article; zbMATH DE number 881246