#### Volume 10, issue 1 (2006)

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Distortion in transformation groups

### Appendix: Yves de Cornulier

Geometry & Topology 10 (2006) 267–293
 arXiv: math.DS/0509701
##### 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 $Homeo\left({\mathbf{S}}^{n}\right)$, thought of as a discrete group.

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

##### Keywords
distortion, transformation groups, Pixton action, Bergman property
##### Mathematical Subject Classification 2000
Primary: 37C85
Secondary: 37C05, 22F05, 57S25, 57M60
##### Publication
Accepted: 8 February 2006
Published: 26 March 2006
Proposed: Benson Farb
Seconded: Leonid Polterovich, Robion Kirby
##### Authors
 Danny Calegari Department of Mathematics California Institute of Technology Pasadena CA 91125 USA Michael H Freedman Microsoft Research 1 Microsoft Way Redmond WA 98052 USA Yves de Cornulier Institut de Mathématiques Université de Neuchâtel Rue Émile Argand 11 CH-2007 Neuchâtel Switzerland