Strong distortion in transformation groups

Abstract: We discuss boundedness and distortion in transformation groups. We show that the groups $\mathrm{Diff}^r_0(\mathbb{R}^n)$ and $\mathrm{Diff}^r(\mathbb{R}^n)$ have the strong distortion property, whenever $0 \leq r \leq \infty, r \neq n+1$. 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 $\mathrm{Diff}_0^r(M)$ satisfy a relative Higman embedding type property, introduced by Schreier. This answers a problem asked by Schreier in the famous Scottish Book.