Maximal pseudometrics and distortion of circle diffeomorphisms

Abstract: We initiate a study of distortion elements in the Polish groups $\mbox{Diff}<em>+<sup>k(\mathbb{S}<sup>1)$ ($1\leq k&lt;\infty$), as well as $\mbox{Diff}</em>+<sup>{1+AC}(\mathbb{S}<sup>1)$, in terms of maximal metrics on these groups. We classify distortion in the $k=1$ case: a $C<sup>1$ circle diffeomorphism is $C<sup>1$-undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms with only non-hyperbolic fixed points which are $C<sup>{1+AC}$-undistorted, and hence $C<sup>k$-undistorted for all $k\geq 2$. In the appendix, we exhibit a maximal metric on $\mbox{Diff}_+<sup>{1+AC}(\mathbb{S}<sup>1)$, and observe that this group is quasi-isometric to a hyperplane of $L<sup>1(I)$.