Smooth perfectness for the group of diffeomorphisms

Stefan Haller; Tomasz Rybicki; Josef Teichmann

doi:10.3934/jgm.2013.5.281

Article Contents

2013, Volume 5, Issue 3: 281-294. Doi: 10.3934/jgm.2013.5.281

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Smooth perfectness for the group of diffeomorphisms

1.
Department of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Vienna
2.
Faculty of applied mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow
3.
Department of Mathematics, ETH Zürich, Rämistrasse 10, 8092 Zürich

Received: February 28, 2013

Published: August 2013

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Abstract

Given a result of Herman, we provide a new elementary proof of the fact that the connected component of the identity in the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism $g$, which is sufficiently close to the identity, can be represented as a product of four commutators, $g=[h_1,k_1]\circ\cdots\circ[h_4,k_4]$, where the factors $h_i$ can be chosen to depend smoothly on $g$, and $k_i=\exp(X_i)$ are flows at time one of complete vector fields $X_i$ which are independent of $g$.

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Mathematics Subject Classification: 58D05.

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References

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