On the hyperbolic limit points of groups acting on hyperbolic spaces
type: Item
instance of: scholarly article
title: On the hyperbolic limit points of groups acting on hyperbolic spaces
zbMATH Open document ID: 0904.20028
DOI: 10.1007/BF02844721
author: Marco Pavone
publication date: 1998-10-01T00:00:00Z
review text: This paper is mainly a survey on the notion of hyperbolic limit point in a few situations of groups acting on hyperbolic spaces. If \(X\) is a metric space, \(\widehat X\) a compactification of \(X\) and \(G\) a group of automorphisms of \(X\) inducing (in an appropriate sense) an action on \(\widehat X\), then the author says that \(g\in G\) is \textit{hyperbolic} if there exist two distinct points \(g^+\) and \(g^-\) in \(\partial X=\widehat X\setminus X\) such that \(g\) moves the points in \(X\) towards \(g^+\) and away from \(g^-\). The author is interested in the following question: under which conditions does there exist hyperbolic elements \(g\) and \(h\in G\) such that \(g^+=h^+\) but \(g^-\not=h^-\). He studies the question particularly in the cases where the space \(X\) is the Poincaré disc with \(G\) its group of automorphisms, and where \(X\) is the Cayley graph of a finitely-generated word hyperbolic group \(G\). The author considers also the map \(g^+\mapsto g^-\), defined on the space of fixed points (at infinity) of hyperbolic elements in a word hyperbolic group. He shows that this map is discontinuous everywhere. The author presents finally an open problem on Cayley graphs which generalizes the results that he describes for word hyperbolic groups.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 1172073
zbMATH Keywords: hyperbolic limit points
zbMATH Keywords: compactifications
zbMATH Keywords: boundaries
zbMATH Keywords: automorphism groups
zbMATH Keywords: Cayley graphs
zbMATH Keywords: homogeneous trees
zbMATH Keywords: Poincaré discs
zbMATH Keywords: finitely generated groups
zbMATH Keywords: hyperbolic elements
zbMATH Keywords: word hyperbolic groups
zbMATH Keywords: groups acting on hyperbolic spaces
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1172073