Homomorphisms between diffeomorphism groups
Kathryn Mann

TL;DR
This paper classifies all nontrivial actions of certain diffeomorphism groups on 1-manifolds, revealing they are topologically diagonal, and answers a question about the dimensionality of manifolds admitting such group embeddings.
Contribution
It provides a complete classification of group actions of Diff^r_c(R) and Diff^r_+(S1) on lines and circles, introducing the concept of topologically diagonal actions and addressing manifold dimension questions.
Findings
All actions are topologically diagonal.
Any injective homomorphism from Diff(M)_0 to a 1-manifold diffeomorphism group implies M is 1-dimensional.
Several new lemmas on subgroup structures of 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.
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