Algebraic properties of the group of germs of diffeomorphisms

TL;DR

This paper explores the algebraic structure of the group of germs of analytic diffeomorphisms of complex n-space, revealing properties like residual finiteness, Hopfianity, and automorphism groups, with implications for group embedding constraints.

Contribution

It provides new algebraic insights into the structure and automorphisms of the groups of germs of diffeomorphisms, including their residual finiteness and non-co-Hopfian properties.

Findings

Finitely generated subgroups are residually finite.

The group of formal diffeomorphisms is Hopfian.

The groups are not co-Hopfian.

Abstract

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