, Volume 9, Issue 1, pp 1-28

Patterson-Sullivan Theory in Higher Rank Symmetric Spaces

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Abstract.

Let X = G/K be a Riemannian symmetric space of noncompact type and \( \Gamma \) a discrete “generic” subgroup of G with critical exponent \( \delta(\Gamma) \) . Denote by \( X_{reg} (\infty) \) the set of regular elements of the geometric boundary \( X(\infty) \) of X. We show that the support of all \( \Gamma \) -invariant conformal densities of dimension \( \delta(\Gamma) \) on \( X_{reg} (\infty) \) (e.g. Patterson-Sullivan densities) lie in a same and single regular G-orbit on \( X(\infty) \) . This provides information on the large-scale growth of \( \Gamma \) -orbits in X. If in addition we assume \( \Gamma \) to be of divergence type, then there is a unique density of the previous type. Furthermore, we explicitly determine \( \delta(\Gamma) \) and this G-orbit for lattices, and show that they are of divergence type.

Submitted: November 1997, revised: January 1999.