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

We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 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 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