Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

Abstract

We construct finitely generated groups with arbitrary prescribed Hilbert space compression \alpha from the interval [0,1]. 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 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.Comment: 21 pages; version 3: The final version, accepted by Crelle; version 2: corrected misprints, added references, the group has asdim at most 2, not at most 3 as in the first version (thanks to A. Dranishnikov); version 3: took into account referee remarks, added references. the paper is accepted in Crell