# Commentarii Mathematici Helvetici

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**Volume 81, Issue 4, 2006, pp. 911–929**

**DOI: 10.4171/CMH/80**

Published online: 2006-12-31

Metrics on diagram groups and uniform embeddings in a Hilbert space

Goulnara N. Arzhantseva^{[1]}, V. S. Guba

^{[2]}and Mark V. Sapir

^{[3]}(1) Universität Wien, Austria

(2) Vologda State University, Russian Federation

(3) Vanderbilt University, Nashville, United States

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 $\mathbb{Z}\wr H$.

*Keywords: *Richard Thompson's group $F$, diagram groups, Hilbert space compression, subgroup distortion

Arzhantseva Goulnara, Guba V., Sapir Mark: Metrics on diagram groups and uniform embeddings in a Hilbert space. *Comment. Math. Helv.* 81 (2006), 911-929. doi: 10.4171/CMH/80