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

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