Quasiconvexity of the fixed-point set of an automorphism of a hyperbolic group
Let \(G\) be a word hyperbolic group, i.e., \(G\) is finitely generated and the geodesic triangles in the Cayley graph of \(G\) are \(\delta\)-thin. Let \(\phi\) be an automorphism of \(G\). It was shown by \textit{W.~D.~Neumann} [Invent. Math. 110, No. 1, 147-150 (1992; Zbl 0793.20033)] that the fixed group \(G^\phi\), consisting of the fixed points of \(\phi\), is quasiconvex, i.e., there exists a constant \(c\) such that every geodesic in the Cayley graph of \(G\), joining points of \(G^\phi\), lies in the \(c\)-neighborhood of \(G^\phi\). In the article under review a new proof of this fact is given. The proof uses the Gromov boundary of a hyperbolic group.
scientific article; zbMATH DE number 1432184