Representations of Currents Taking Values in a Totally Disconnected Group

Abstract

Let $G={\mathcal{A}ut(\mathcal{T})}$ be the group of all automorphisms of a homogeneous tree $\mathcal{T}$ of degree q + 1 ≥ 3 and (X, m) a compact metrizable measure space with a probability measure m. We assume that μ has no atoms. The group $\mathcal{G}={\mathcal{A}ut(\mathcal{T})}^X=G^X$ of bounded measurable currents is the completion of the group of step functions $f:X\to{\mathcal{A}ut(\mathcal{T})}$ with respect to a suitable metric. Continuos functions form a dense subgroup of $\mathcal{G}$ . Following the ideas of I.M. Gelfand, M.I. Graev and A.M. Vershik we shall construct an irreducible family of representations of $\mathcal{G}$ . The existence of such representations depends deeply from the nonvanisching of the first cohomology group $H^1({\mathcal{A}ut(\mathcal{T})},\pi)$ for a suitable infinite dimensional π.