Maximal pseudometrics and distortion of circle diffeomorphisms

We initiate a study of distortion elements in the Polish groupsDiff+k(S1)(1 <= k<infinity), as well asDiff+1+AC(S1), in terms of maximal metrics on these groups. We classify distortion in thek=1case: aC1circle diffeomorphism isC1-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 nonhyperbolic fixed points which areC1+AC-undistorted, and henceCk-undistorted for allk > 2. In the Appendix, we exhibit a maximal metric onDiff+1+AC(S1), and observe that this group is quasi-isometric to a hyperplane ofL1(I).