Ergodic properties of discrete groups; inheritance to normal subgroups and invariance under quasiconformal deformations (Q1261744)

The author gives some results concerning ergodic properties of isometry groups \(\Gamma\) acting on the \(n\)-dimensional hyperbolic space \(B^ n=\{x\in\mathbb{R}^ n\bigl|\) \(| x|<1\}\) and on the sphere \(S^{n-1}\). These results are in connection with those obtained by \textit{T. Lyons} and \textit{D. Sullivan} [J. Diff. Geom. 19, 299-323 (1984; Zbl 0554.58022)]. In the paper under review it is studied in what degree any normal subgroup of \(\Gamma\) inherits ergodicity on \(S^{n-1}\times S^{n-1}\) and on \(S^{n-1}\). In the case \(n=2\) it is proved that \(B^ 2/\Gamma\) is recurrent iff any nontrivial normal subgroup of \(\Gamma\) is conservative. It is studied whether the ergodic properties on \(S^{n-1}\) are preserved or not by deformations of \(\Gamma\) under quasi-isometric automorphisms of \(B^ n\).