The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L ₂ and a constant c > 0 such that for all x,y ∈ G we have ||f(x) - f(y)|| ₂ ≥ cd(x,y) ^α. It was previously known that the Hilbert compression exponent of the wreath product ℤ ∼ ℤ is between 2/3 and 3/4. Here we show that 2/3 is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.