Homomorphisms between diffeomorphism groups

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.