Homomorphisms between diffeomorphism groups
Abstract
For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial 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 Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 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.
Cite
@article{arxiv.1206.1196,
title = {Homomorphisms between diffeomorphism groups},
author = {Kathryn Mann},
journal= {arXiv preprint arXiv:1206.1196},
year = {2013}
}
Comments
Contains corrections and additional references. A revised version will appear in Ergodic Theory and Dynamical Systems