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

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.