Recurrence in unipotent groups and ergodic nonabelian group extensions

Let T be a measure-preserving and ergodic transformation of a standard probability space (X, 8, mu) and let f : X -> SUTd(R) be a Borel map into the group of unipotent upper triangular d x d matrices. We modify an argument in [12] to obtain a sufficient condition for the recurrence of the random walk defined by f, in terms of the asymptotic behaviour of the distributions of the suitably scaled maps f (n, x) = (fT(n-1 .) fT(n-2...) fT (.) f). We give examples of recurrent cocycles with values in the continuous Heisenberg group H-1(R) = SUT3(R), and we use a recurrent cocycle to construct an ergodic skew-product extension of an irrational rotation by the discrete Heisenberg group H-1(Z) = SUT3(Z).