Free group representations from vector-valued multiplicative functions, I

Let G be the group of all automorphisms of a homogeneous tree T of degree greater than three and (X,m)a compact metrizable measure space with a probability measure m. We assume that m has no atoms. The group G^X of bounded measurable currents is the completion of the group of step functions with respect to a suitable metric. Continuos functions form a dense subgroup. Following the ideas of I.M. Gelfand, M.I. Graev and A.M. Vershik we shall construct an irreducible family of representations of G^X. The existance of such representations depends deeply from the nonvanisching of the first cohomology group H1(G,p) for a suitable infinite dimentional p.