The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Abstract

Let G denote a semisimple group, Γ a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following: (1) For any Γ-quasi-invariant measure η on B, and any probablity measure μ on Γ, the norm of the operator πη(μ) on L 2(B,η) is equal to ∥λΓ(μ)∥, where πη is the unitary representation in L 2(X,η), and λΓ is the regular representation of Γ. (2) In particular this estimate holds when η is Lebesgue measure on B, a Patterson–Sullivan measure, or a μ-stationary measure, and implies explicit lower bounds for the displacement and Margulis number of Γ (w.r.t. a finite generating set), the dimension of the conformal density, the μ-entropy of the measure, and Lyapunov exponents of Γ. (3) In particular, when G=PSL2(ℂ) and Γ 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 ∥πη(μ)∥=∥λG(μ)∥ for any amenable action of G and μ∈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 μ-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 Γ is any word-hyperbolic group, acting on their Poisson boundary, for example.