Algebraic properties of groups of complex analytic local diffeomorphisms

Abstract

We study in these notes the algebraic properties of groups of holomorphic local diffeomorphisms. In this spirit we introduce the basic notions of the theory of pro-algebraic groups. Pro-algebraic groups are the analogue of algebraic linear groups in the infinite dimensional setting of groups of holomorphic local diffeomorphisms. They are very useful to study properties that determine groups defined by algebraic equations in every space of jets.