Entropies and Poisson boundaries of random walks on groups with rapid decay

Abstract

Let $G$ be a countable group and $μ$ a probability measure on $G$. We build a new framework to compute asymptotic quantities associated with the $μ$-random walk on $G$, using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the $μ$-random walk with respect to a weight $ω$ on $G$ can be computed in terms of the asymptotic behavior of the spectral radius of $μ$ in an ascending class of weighted group algebras, and we prove that for natural choices of $ω$ and $μ$, the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the $μ$-random walk can be realized as the Lyapunov exponent of $μ$ with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of $p$-pseudofunctions on $G$, we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.