  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.
