Strong distortion in transformation groups

We discuss boundedness and distortion in transformation groups. We show that the groups Diff0(R) and Diff (Rn) have the strong distortion property, whenever 0 ≤ r ≤ ∞, r 6= n+ 1. This implies in particular that every abstract length function on these groups is bounded. With related techniques we show that, for M a closed manifold or homeomorphic to the interior of a compact manifold with boundary, the groups Diff0(M) satisfy a relative Higman embedding type property, introduced by Schreier. This answers a problem asked by Schreier in the famous Scottish Book.