Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

TL;DR

This paper investigates the distribution and equidistribution of lattice orbits on symmetric spaces and their boundaries, providing new insights into orbit behavior and applications to Patterson-Sullivan theory.

Contribution

It establishes the equidistribution of lattice orbits on the Furstenberg boundary and in sectors of symmetric spaces, using the strong wavefront lemma and solvable flow equidistribution.

Findings

Orbits of  in the Furstenberg boundary are equidistributed.

Orbits of  in the symmetric space are equidistributed in sectors.

Application to Patterson-Sullivan theory enhances understanding of limit sets.

Abstract

Let X be a symmetric space of noncompact type and \Gamma a lattice in the isometry group of X. We study the distribution of orbits of \Gamma acting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in X and b in X(\infty), we investigate the distribution of the set {(y\gamma,b\gamma^{-1}):\gamma\in\Gamma} in X\times X(\infty). It is proved, in particular, that the orbits of \Gamma in the Furstenberg boundary are equidistributed, and that the orbits of \Gamma in X are equidistributed in ``sectors'' defined with respect to a Cartan decomposition. We also discuss an application to the Patterson-Sullivan theory. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces.