Algebraic Properties of the Group of Germs of Diffeomorphisms

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