Strong distortion in transformation groups

We discuss boundedness and distortion in transformation groups. We show that the groups ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0^r({\mathbb{R}}^n)$ and ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^r({\mathbb{R}}^n)$ have the strong distortion property, whenever $0\le r\le \mathrm{\infty },r\mathrm{\ne }n+1$. 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 ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0^r(M)$ satisfy a relative Higman embedding type property, introduced by Schreier. This answers a problem asked by Schreier in the famous Scottish Book.