  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.
