New proofs of theorems of Kathryn Mann

TL;DR

This paper presents shorter proofs for two theorems by Kathryn Mann concerning the rigidity of certain diffeomorphism groups, highlighting their structural properties and actions on the circle.

Contribution

The paper provides more concise proofs of Mann's theorems on the non-existence of certain group actions and the characterization of endomorphisms of diffeomorphism groups.

Findings

The identity component of compactly supported diffeomorphisms of R^n cannot act nontrivially on S^1 for n≥2.

Nontrivial endomorphisms of the diffeomorphism group of S^1 are conjugations by C^r diffeomorphisms for r≥3.

Abstract

We give a shorter proof of the following theorem of Kathryn Mann \cite{M}: the identity component of the group of the compactly supported $C^r$ diffeomorphisms of $\R^n$ cannot admit a nontrivial $C^p$-action on $S^1$, provided $n\geq2$, $r\neq n+1$ and $p\geq2$. We also give a new proof of another theorem of Mann: any nontrivial endomorphism of the group of the orientation preserving $C^r$ diffeomorphisms of the circle is the conjugation by a $C^r$ diffeomorphism, if $r\geq3$.