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Homomorphisms between diffeomorphism groups

Published online by Cambridge University Press:  04 July 2013

KATHRYN MANN
KATHRYN MANN*
Affiliation:
Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637, USA email mann@math.uchicago.edu
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Abstract

For $r\geq 3$, $p\geq 2$, we classify all actions of the groups ${ \mathrm{Diff} }_{c}^{r} ( \mathbb{R} )$ and ${ \mathrm{Diff} }_{+ }^{r} ({S}^{1} )$ by ${C}^{p} $-diffeomorphisms on the line and on the circle. This is the same as describing all non-trivial group homomorphisms between groups of compactly supported diffeomorphisms on 1-manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if $M$ is any closed manifold, and ${\mathrm{Diff} }^{\infty } \hspace{-2.0pt} \mathop{(M)}\nolimits_{0} $ injects into the diffeomorphism group of a 1-manifold, must $M$ be one-dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 1 , February 2015 , pp. 192 - 214
Copyright
© Cambridge University Press, 2013 

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