 
Summary: J. reine angew. Math. aa (2009), 123
DOI 10.1515/CRELLE.2009.aaa
Journal fu¨r die reine und
angewandte Mathematik
( Walter de Gruyter
Berlin Á New York 2009
Compression functions of uniform embeddings of
groups into Hilbert and Banach spaces
By Goulnara Arzhantseva at Gene`ve, Cornelia Drut¸u at Oxford,
and Mark Sapir at Nashville
Abstract. We construct finitely generated groups with arbitrary prescribed Hilbert
space compression a A ½0; 1. This answers a question of E. Guentner and G. Niblo.
For a large class of Banach spaces E (including all uniformly convex Banach spaces), the
Ecompression of these groups coincides with their Hilbert space compression. Moreover,
the groups that we construct have asymptotic dimension at most 2, hence they are exact. In
particular, the first examples of groups that are uniformly embeddable into a Hilbert space
(moreover, of finite asymptotic dimension and exact) with Hilbert space compression 0 are
given. These groups are also the first examples of groups with uniformly convex Banach
space compression 0.
1. Introduction
