Strong distortion in transformation groups

We show that the groups Diff0r(Rn) and Diffr(Rn) have the strong distortion property, whenever 0⩽r⩽∞,r≠n+1. This implies in particular that every element in these groups is distorted, a property with dynamical implications. The result also gives new examples of groups with Bergman's strong boundedness property as in Bergman (Bull. Lond. Math. Soc. 38 (2006) 429–440). With related techniques we show that, for M a closed manifold or homeomorphic to the interior of a compact manifold with boundary, the diffeomorphism groups Diff0r(M) satisfy a relative Higman embedding type property, introduced by Schreier. In the simplest case, this answers a problem asked by Schreier in the famous Scottish Book.