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Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. (English) Zbl 1132.22012

Assume that \(X\) is a symmetric space of noncompact type, and denote by \(X(\infty )\) its geometric boundary. Moreover, let \(G\) be the identity component of the isometry group of \(X\) acting on \(X\). Let \(\Gamma\) be a lattice in \(G\), i.e.a discrete subgroup with finite covolume. One can extend the action of \(\Gamma\) on \(X\) to \(X(\infty )\). Given \(y\in X\) and \(b\in X(\infty )\) the distribution of \(\{(y\gamma ,b\gamma^{-1}):\gamma\in\Gamma\}\) is investigated. It is proved that the orbits of \(\Gamma\) in the Furstenberg boundary are equidistributed. Furthermore, the orbits of \(\Gamma\) in \(X\) are equidistributed in “sectors” defined with respect to a Cartan decomposition.
Reviewer: Peter Raith (Wien)

MSC:

22E40 Discrete subgroups of Lie groups
37A17 Homogeneous flows
22F30 Homogeneous spaces
37A15 General groups of measure-preserving transformations and dynamical systems
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