Distortion in transformation groups

Abstract

We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S${}^n$, thought of as a discrete group.

An appendix by Y de Cornulier shows that Homeo(S${}^n$ has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(S${}^n$ on a metric space by isometries has bounded orbits.