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Distortion in transformation groups (with an appendix by Yves de Cornulier). (English) Zbl 1106.37017
Summary: We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J. Franks and M. Handel [Duke Math. J. 131, No. 3, 441–468 (2006; Zbl 1088.37009)]. 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.
The appendix by Y. de Cornulier shows that Homeo$$(S^n)$$ has the strong boundedness property, recently introduced by G. Bergman [Bull. Lond. Math. Soc. 38, No. 3, 429-440 (2006; Zbl 1103.20003 )]. This means that every action of the discrete group Homeo$$(S^n)$$ on a metric space by isometries has bounded orbits.

##### MSC:
 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 57M60 Group actions on manifolds and cell complexes in low dimensions 57S25 Groups acting on specific manifolds 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 22F05 General theory of group and pseudogroup actions
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