The solvable length of groups of local diffeomorphisms

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 $n=2,3,4,5$.