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

2004/05/27 by Alexander Gorodnik, Hee Oh, Gorodnik, Alexander +1 · 2 citations
Mathematics · #22E40 #37A17 #57S30 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #Mathematical Dynamics and Fractals #math.DS #math.GR #msc:22E40 #msc:37A17 #msc:57S30

paper · pdf · doi:10.48550/arxiv.math/0405515

arxiv created 2004/05/27 · openalex publication_date 2004/05/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Let X be a symmetric space of noncompact type and Γa lattice in the isometry group of X. We study the distribution of orbits of Γacting on the symmetric space X and its geometric boundary X(∞). More precisely, for any y in X and b in X(∞), we investigate the distribution of the set (yγ,bγ^-1):γ∈Γ in X× X(∞). It is proved, in particular, that the orbits of Γin the Furstenberg boundary are equidistributed, and that the orbits of Γin 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.

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• Counting totally real units and eigenvalue patterns in \rmSL_n(\mathbb Z) and \rmSp_2n(\mathbb Z) in thin tubes
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