Almost block independence for the three dot ℤ² dynamical system

We show that the measure preserving action of Z2 dual to the action defined by the commuting automorphisms ×x and ×y on the discrete group Z[x±1,y±1]/á1+x+yñZ[x±1,y±1] is measurably isomorphic to a Z2 Bernoulli shift. This was conjectured in recent work by Lind, Schmidt and the author, where it was shown that this action has completely positive entropy. An example is given of Z2 actions which are measursbly isomorphic without being topologically conjugate.