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