Distortion of imbeddings of groups of intermediate growth into metric spaces

Abstract:

We show that groups of subexponential growth can have arbitrarily bad distortion for their imbeddings into Hilbert space.

More generally, consider a metric space $\mathcal X$, and assume that it admits a sequence of finite groups of bounded-size generating set which does not imbed coarsely in $\mathcal X$. Then, 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 implies that Liouville groups may have arbitrarily bad distortion for their imbeddings into Hilbert space, precluding a converse to the result by Naor and Peres that groups with distortion much better than $\sqrt t$ are Liouville.