Metrics on diagram groups and uniform embeddings in a Hilbert space

Abstract

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 .