Algebraic properties of the group of germs of diffeomorphisms

We establish some algebraic properties of the group $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^n,0)$ of germs of analytic diffeomorphisms of ${\mathbb{C}}^n$, and its formal completion $\widehat{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}({\mathbb{C}}^n,0)$. For instance we describe the commutator of $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^n,0)$, but also prove that any finitely generated subgroup of $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^n,0)$ is residually finite; we thus obtain some constraints of groups that embed into $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^n,0)$. We show that $\widehat{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}({\mathbb{C}}^n,0)$ is an Hopfian group, and that $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^n,0)$ and $\widehat{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}({\mathbb{C}}^n,0)$ are not co-Hopfian. We end by the description of the automorphisms groups of $\widehat{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}(\mathbb{C},0)$, and $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(\mathbb{C},0)$.