  Let X and Y be infinite graphs, such that the automorphism group of X is
nonamenable, and the automorphism group of Y has an infinite orbit. We prove
that there is no automorphism-invariant measure on the set of spanning trees in
the direct product X times Y. This implies that the minimal spanning forest
corresponding to i.i.d. edge-weights in such a product, has infinitely many
connected components almost surely.
