Algebraic convergence of function groups
type: Item
instance of: scholarly article
title: Algebraic convergence of function groups
zbMATH Open document ID: 1008.30026
author: Gero Kleineidam
author: Juan Souto
published in: Commentarii Mathematici Helvetici
publication date: 2002-09-25T00:00:00Z
review text: In this article the authors consider a non-elementary torsion-free geometrically finite function group \(\Gamma_{0}\), which is not a quasifuchsian one, without parabolic transformations and so that any quasifuchian component is of genus \(2\) (convex co-compact). If we denote by \(\Omega\) the region of discontinuity of \(\Gamma_{0}\), then there is a unique invariant component \(\Omega_{0}\) and a finite collection of non-invariant ones \(\Omega_{1}\),..., \(\Omega_{s}\), which are non-equivalents, so that \(\Omega/\Gamma_{0}=\Omega_{0}/\Gamma_{0} + \Omega_{1}/\Gamma_{1} + \cdots + \Omega_{s}/\Gamma_{s}\), where \(\Gamma_{j}\) is the stabilizer of the component \(\Omega_{j}\) in \(\Gamma_{0}\). The quasiconformal deformation space \(QH(\Gamma_{0})\) is an open subset of \(D(\Gamma_{0})\), the deformation space of \(\Gamma_{0}\). The universal covering of \(QH(\Gamma_{0})\) is given by the product of the Teichmüller spaces \({\mathcal T}_{j}\) of the surfaces \(S_{j}=\Omega_{j}/\Gamma_{j}\), for \(j=0,1,..,s\). The non-compactness of \(QH(\Gamma_{0})\) in \(D(\Gamma_{0})\) makes natural to ask when a divergent sequence in \(QH(\Gamma_{0})\) converges in \(D(\Gamma_{0})\). R.D.Canary has shown that if \(K \subset {\mathcal T}_{0}\) is compact, then its image in \(QH(\Gamma_{0})\) has compact closure in \(D(\Gamma_{0})\). The authors consider the case of sequences in \(QH(\Gamma_{0})\) for which the corresponding sequence in \({\mathcal T}_{0}\) diverges. A conjecture due to W.P. Thurston asserts that such a sequence will converge in \(D(\Gamma_{0})\) if the corresponding sequence in \({\mathcal T}_{0}\) has a convergent subsequence in the Thurston's compactification given by projective measured laminations of \(S_{0}\). There is an open subset of such a compactification, called the Masur domain, on which the extended action of the mapping class group acts discontinuously. The main two results of the authors asserts that (i) if the sequence in \(QH(\Gamma_{0})\) has …
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1807860
zbMATH Keywords: hyperbolic manifolds
zbMATH Keywords: compression bodies
zbMATH Keywords: algebraic convergence
zbMATH Keywords: Masur domain
zbMATH Keywords: \(R\)-trees
reviewed by: Rubén A. Hidalgo
MaRDI profile type: Publication
full work available at URL: s00014-002-8339-x
OpenAlex ID: W2045568052
DOI: 10.1007/S00014-002-8339-X
description: scientific article; zbMATH DE number 1807860