A zoo of diffeomorphism groups on $\mathbb R^n$

TL;DR

This paper studies various groups of smooth diffeomorphisms on b^n that differ from the identity by functions with specific decay or boundedness properties, proving they are all smooth regular Lie groups.

Contribution

It establishes that these different classes of diffeomorphism groups are all smooth regular Lie groups, expanding understanding of their structure and properties.

Findings

All considered diffeomorphism groups are smooth regular Lie groups.

The groups differ by functions in b, H^, or b with specific decay/boundedness.

The results unify various diffeomorphism groups under the framework of smooth regular Lie groups.

Abstract

We consider the groups $\operatorname{Diff}_{\mathcal B}(\mathbb R^n)$, $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$, and $\operatorname{Diff}_{\mathcal S}(\mathbb R^n)$ of smooth diffeomorphisms on $\mathbb R^n$ which differ from the identity by a function which is in either $\mathcal B$ (bounded in all derivatives), $H^\infty = \bigcap_{k\ge 0}H^k$, or $\mathcal S$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.