Metrics on diagram groups and uniform embeddings in a Hilbert space

We give first examples of finitely generated groups having an intermediate, with values in (0, 1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson's group F is equal to 1/2, the Hilbert space compression of Z{Z is between 1/2 and 3/4, and the Hilbert space compression of Z{(Z{Z) is between 0 and 1/2. In general, we find a relationship between the growth of H and the Hilbert space compression of Z{H.