On the hyperbolic limit points of groups acting on hyperbolic spaces
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.
scientific article; zbMATH DE number 1172073
scientific article