Algebraic properties of the group of germs of diffeomorphisms

Abstract

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