 
Summary: Comment. Math. Helv. 81 (2006), 911929 Commentarii Mathematici Helvetici
© Swiss Mathematical Society
Metrics on diagram groups and uniform embeddings in a Hilbert
space
G. N. Arzhantseva, V. S. Guba and M. V. Sapir
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 Z Z is between 1/2 and 3/4, and the
Hilbert space compression of Z (Z 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 H.
Mathematics Subject Classification (2000). 20F65, 20F69
Keywords. Richard Thompson's group F, diagram groups, Hilbert space compression, sub
group distortion.
1. Introduction
The study of uniform embeddings of metric spaces into Hilbert space was initiated
by Gromov.
Definition 1.1. Let ( , d) be a metric space. Let H be a separable Hilbert space. A
map f : H is said to be a uniform embedding [9] if there exist nondecreasing
