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 E-compression 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

more »

... 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. This raised the question whether every finitely generated group can be embedded uniformly into a Hilbert space, or more generally, into a uniformly convex Banach space. Gromov constructed [Gr2] finitely generated random groups whose Cayley graphs (quasi)contain some infinite families of expanders and thus cannot be embedded uniformly into a Hilbert space (or into any l p with 1 e p < y, e.g. [Roe], Ch.11.3). The recent results of V. La¤orgue [Laf ] yield a family of expanders that is not uniformly embeddable into any This article was written while all the three authors were visitors at the Max Planck Institute in Bonn. We are grateful to the MPI for its hospitality. A very limited information was known about the possible values of Hilbert space compression of finitely generated groups. For example, word hyperbolic groups have Hilbert space compression 1 [BS], and so do groups acting properly and co-compactly on a CAT(0) cube complex [CN]; co-compact lattices in arbitrary Lie groups, and all lattices in semi-simple Lie groups have Hilbert space and, moreover, L p -compression 1 [Te]; any group that is not uniformly embeddable into a Hilbert space (such groups exist by [Gr2]) has Hilbert space compression 0, etc. (see the surveys in [AGS], [Te]). The first groups with Hilbert space compressions strictly between 0 and 1 were found in [AGS]: R. Thompson's group F has Hilbert space compression 1=2, the Hilbert space compression of the wreath product Z o Z is between 1=2 and 3=4 (later it was proved in [ANP] that it is actually 2=3), the Hilbert space compression of Z o ðZ o ZÞ is between 0 and 1=2. 214 Arzhantseva, Druţu and Sapir, Compression functions of uniform embeddings of groups