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We show that groups of subexponential growth can have arbitrarily bad distortion for their imbeddings into Hilbert space. More generally, consider a metric space Χ, and assume that it admits a sequence of finite groups of bounded-size generating set which does not imbed coarsely in Χ. Then, for every unbounded increasing function ρ, we produce a group of subeΧponential word growth all of whose imbeddings in Χ have distortion worse than ρ. 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 √t are Liouville. [ABSTRACT FROM AUTHOR]