Exponential decay of matrix coefficients.

For a simple Lie group G and a discrete subgroup Gamma of G, we will discuss  exponential decay of the matrix coefficients of G associated to functions on the quotient space Gamma\G. When G is of rank one, and Gamma is a lattice, this yields exponential mixing of the geodesic flow for  the associated Riemannian manifold.  We will also discuss a uniform  exponential decay for congruence subgroups of Gamma when Gamma is contained in an arithmetic subgroup of G.

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