  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 the restricted wreath product $Z\wr Z$ is between
1/2 and 3/4, and the Hilbert space compression of $Z\wr (Z\wr Z)$ is between 0
and 1/2. In general, we find a relationship between the growth of $H$ and the
Hilbert space compression of $Z\wr H$.
