Finite-dimensional representations constructed from random walks

Let an amenable group G and a probability measure \mu on it (that is finitely-supported, symmetric, and non-degenerate) be given. I will present a construction, via the \mu-random walk on G, of a harmonic cocycle and the associated orthogonal representation of G. Then I describe when the constructed orthogonal representation contains a non-trivial finite-dimensional subrepresentation (and hence an infinite virtually abelian quotient), and some sufficient  conditions for G to satisfy Shalom's property HFD. (joint work with A. Erschler, arXiv:1609.08585)