Distortion of imbeddings of groups of intermediate growth into metric   spaces
  For every metric space $\mathcal X$ in which there exists a sequence of finite groups of bounded-size generating set that does not embed coarsely, and for every unbounded, increasing function $\rho$, we produce a group of subexponential word growth all of whose imbeddings in $\mathcal X$ have distortion worse than $\rho$.   This applies in particular to any B-convex Banach space $\mathcal X$, such as Hilbert space. 
Laurent Bartholdi, Anna G. Erschler
math.GR