Algebraic convergence of finitely generated Kleinian groups in all dimensions
type: Item
instance of: scholarly article
title: Algebraic convergence of finitely generated Kleinian groups in all dimensions
zbMATH Open document ID: 1188.30054
author: Shihai Yang
published in: Linear Algebra and its Applications
publication date: 2010-02-12T00:00:00Z
review text: Let \(\{G_{i,r}\}\), \(G_{i,r}=\langle g_{11},\dots, g_{1r}\rangle\), be a sequence of \(r\)-generator subgroups of the \(n\)-dimensional orientation preserving Möbius group such that for each \(k\in \{1,\dots, r\}\) the sequence \(\{g_{ik}\}\) converges to a Möbius transformation \(g_k\). Then we say that \(\{G_{i,r}\}\) converges algebraically to the group \(G= \langle g_1,\dots, g_r\rangle\). We are interested in the case that all \(G_{i,r}\) are Kleinian groups, that is, all \(G_{i,r}\) are discrete and non-elementary. The question arises now under what conditions also the algebraic limes \(G\) is a Kleinian group. If \(\{G_{i,r}\}\) has uniformly bounded torsion, that is, if there exists an integer \(N\geq 1\) such that \(\text{ord}(g)\leq N\) or \(\text{ord}(g)=\infty\) for all \(g\in G_{i,r}\) for some \(i\), then \(G\) is indeed a Kleinian group. Here the author extends this result by replacing the hypothesis of uniformly bounded torsion by a weaker and more technical hypothesis which is called Condition A.
reviewed by: Gerhard Rosenberger
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 5669171
zbMATH Keywords: Möbius groups
zbMATH Keywords: Kleinian groups
zbMATH Keywords: algebraic convergence
zbMATH Keywords: uniformly bounded torsion
MaRDI profile type: Publication
full work available at URL: j.laa.2009.10.024
OpenAlex ID: W2063221354
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DOI: 10.1016/J.LAA.2009.10.024
description: scientific article; zbMATH DE number 5669171