 
Summary: 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 com
pression (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 .
Section de Math’ ematiques, Universit’ e de Gen‘ eve, CP 64, 1211 Gen‘ eve
4, Switzerland
Email address : Goulnara.Arjantseva@math.unige.ch
Department of mathematics, Vologda State University, 6 S. Orlov
St., Vologda, 160600, Russia
Email address : guba@univologda.ac.ru
Department of mathematics, Vanderbilt University, USA
