We give first examples of finitely generated groups having an intermediate, with values in , 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 is equal to , the Hilbert space compression of is between and , and the Hilbert space compression of is between 0 and . In general, we find a relationship between the growth of and the Hilbert space compression of .
Cite this article
Goulnara N. Arzhantseva, V. S. Guba, Mark V. Sapir, Metrics on diagram groups and uniform embeddings in a Hilbert space. Comment. Math. Helv. 81 (2006), no. 4, pp. 911–929DOI 10.4171/CMH/80