Nonamenable products are not treeable

Let X and Y be infinite graphs such that the automorphism group of X is non amenable and the automorphism group of Y has an infinite orbit. We prove th at there is no automorphism-invariant measure on the set of spanning trees in the direct product Xx Y. This implies that the minimal spanning forest c orresponding to i.i.d. edge-weights in such a product has infinitely many c onnected components almost surely.