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

We consider the groups ${\mathrm{Diff}}_{\mathcal{B}}({\mathbb{R}}^n)$, ${\mathrm{Diff}}_{H^{\mathrm{\infty }}}({\mathbb{R}}^n)$, and ${\mathrm{Diff}}_{\mathcal{S}}({\mathbb{R}}^n)$ 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), $H^{\mathrm{\infty }}={\bigcap }_{k\ge 0}{H}^k$, or $\mathcal{S}$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.