The solvable length of groups of local diffeomorphisms

Abstract

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 2 ⁢ n + 1 {2n+1} is the sharpest upper bound for the derived length of solvable subgroups of the group Diff ⁢ ( ℂ n , 0 ) {\mathrm{Diff}({\mathbb{C}}^{n},0)} of local complex analytic diffeomorphisms for n = 2 , 3 , 4 , 5 {n=2,3,4,5} .