The spectral theory of amenable actions and invariants of discrete groups

Let G denote a semisimple group, Gamma a discrete subgroup, B = G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following: (1) For any Gamma-quasi-invariant measure eta on B, and any probablity measure mu on Gamma, the norm of the operator pi(eta)(mu) on L-2(B, eta) is equal to parallel tolambda(Gamma)(mu)parallel to, where pi(eta) is the unitary representation in L-2 (X, eta), and lambda(Gamma) is the regular representation of Gamma. (2) In particular this estimate holds when eta is Lebesgue measure on B, a Patterson-Sullivan measure, or a mu-stationary measure, and implies explicit lower bounds for the displacement and Margulis number of Gamma (w.r.t. a finite generating set), the dimension of the conformal density, the mu-entropy of the measure, and Lyapunov exponents of Gamma. (3) In particular, when G = PSL2(C) and Gamma is free, the new lower bound of the displacement is somewhat smaller than the Culler - Shalen bound ( which requires an additional assumption) and is greater than the standard ball-packing bound. We also prove that parallel topi(eta)(mu)parallel to=parallel tolambda(G)(mu)parallel to for any amenable action of G and mu is an element of L-1(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the mu-entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when Gamma is any word-hyperbolic group, acting on their Poisson boundary, for example.