The solvable length of groups of local diffeomorphisms

TL;DR

This paper establishes the maximum derived length of solvable groups of local complex analytic diffeomorphisms in dimensions 2 to 5, revealing a sharp bound related to the ambient space's dimension.

Contribution

It proves that the upper bound of 2n+1 for the derived length of solvable subgroups is sharp for dimensions 2 through 5, advancing understanding of algebraic properties of local diffeomorphism groups.

Findings

Maximum derived length is 2n+1 for n=2,3,4,5.

The bound is proven to be sharp in these dimensions.

Provides insights into the algebraic structure of local biholomorphism groups.

Abstract

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we show that $2n+1$ is the sharpest upper bound for the derived length of solvable subgroups of the group $\mathrm{Diff}({\mathbb C}^{n},0)$ of local complex analytic diffeomorphisms for $n=2,3,4,5$.