Representations of Currents taking values in a totally disconnected group

Let G = Aut(T ) be the group of all automorphisms of a homogeneous tree 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 G = Aut(T )X = GX of bounded measurable currents is the completion of the group of step functions f : X → Aut(T ) with respect to a suitable metric. Continuos functions form a dense subgroup of G. Following the ideas of I.M. Gelfand, M.I. Graev and A.M. Vershikwe shall construct an irreducible family of representations of G. The existence of such representations depends deeply from the nonvanisching of the first cohomology group H1(Aut(T ), α) for a suitable infinite dimensional p. © Springer Science+Business Media B.V. 2010.