Distortion in transformation groups

Creators: Calegari, Danny; Freedman, Michael H.; de Cornulier, Yves

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(Sn), thought of as a discrete group. An appendix by Y de Cornulier shows that Homeo(Sn) has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(Sn) on a metric space by isometries has bounded orbits.

Additional Information

Proposed: Benson Farb; Seconded: Leonid Polterovich, Robion Kirby Accepted: 8 February 2006; Received: 7 October 2005 The first author would like to thank Michael Handel for suggesting the problem which motivated Theorem A, and to thank him and John Franks for reading preliminary versions of this paper, and for making clarifications and corrections. He would also like to thank Daniel Allcock for some useful comments.

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