How (not) to prove (un)distortion for diffeomorphisms of one-manifolds

Abstract

This article addresses the following general question: Given a one-dimensional manifold $M$ and $1 \le r < s \le \infty$, does there exist a $C^s$ orientation preserving compactly supported diffeomorphism of $M$ that is undistorted in the group $\mathrm{Diff}_{c,+}^s(M)$ of such diffeomorphisms while distorted in the bigger group of $C^r$ diffeomorphisms? Interestingly, the answer is known to be positive in the case $(r,s)=(1,2)$ and negative in the case $(r,s)=(2,\infty)$, according to arXiv:2004.07055 and arXiv:2507.13770, respectively. The first part of this note originates from a failed attempt to extend the ideas of arXiv:2004.07055 to the case $(r,s)=(2,3)$. More precisely, in regularities $C^1$ and $C^2$, obstructions to distortion are provided by drifts of cocycles for isometric actions of $\mathrm{Diff}_{c,+}^r(M)$ on Banach spaces for $r=1$ and $r=2$ (namely, the logarithmic and projective derivatives $f\mapsto \log Df$ and $f\mapsto D\log Df$, respectively). On $\mathrm{Diff}_{c,+}^3(M)$, the so-called Liouville cocycle is a natural candidate when looking for new obstructions, but we show that its drift vanishes for $C^2$-distorted diffeomorphisms (and this holds more generally for any "similar" cocycle). This does not rule out the existence of $C^2$-distorted diffeomorphisms that are $C^3$-undistorted. However, at least in the case of the real line, such a diffeomorphism should have very low regularity. Indeed, extending the methods and results of arXiv:2507.13770, in the second part of this article, we show that every compactly supported $C^2$-distorted diffeomorphism of the real line is $C^r$-distorted provided its differentiability class is larger than $C^{2r+2}$.