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.