Derived length of solvable groups of local diffeomorphisms

TL;DR

This paper establishes bounds on the derived length of solvable groups of local complex analytic diffeomorphisms, extending matrix group results to the setting of local diffeomorphisms and providing optimal bounds for specific subclasses.

Contribution

It introduces bounds on the solvable length of such groups based on the dimension, including optimal bounds for connected, unipotent, and nilpotent groups.

Findings

Bound the solvable length by a function of dimension n

Provided the best bounds for connected, unipotent, and nilpotent groups

Extended matrix group results to local diffeomorphisms

Abstract

Let $G$ be a solvable subgroup of the group $\diff{}{n}$ of local complex analytic diffeomorphisms. Analogously as for groups of matrices we bound the solvable length of $G$ by a function of $n$. Moreover we provide the best possible bounds for connected, unipotent and nilpotent groups.