Description of the Zariski-Closure of a Group of Formal Diffeomorphisms

Given a subgroup G of the group of germs of biholomorphisms, or more generally the group of formal diffeomorphisms, we provide a constructive description of the Zariski-closure of G if it is finitely generated. Absent the finite generation hypothesis, we describe a finite codimensional subgroup of the Zariski-closure of G. We give criteria determining whether a subgroup G of the group of germs of biholomorphisms or the group of formal diffeomorphisms has a finite dimensional Zariski-closure in terms of the properties of some relevant subgroups. For instance, when G is virtually solvable, we consider the subgroup \(G_u\) of G consisting of its unipotent elements. In such a case we show that if \(G/G_u\) and \(G_u\) are finitely generated and \(G_u\) is nilpotent then G is finite dimensional. We discuss the geometrical relevance of the Zariski-closure and the finite dimension property and also briefly review part of the algebraic theory of germs of biholomorphisms and some of its last advances.