Description of the Zariski-closure of a group of formal diffeomorphisms (Q6604710)

The paper is most adequately reviewed by quoting the authors abstract:\N\N``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 \(G\).\N\NWe 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.\N\NWe 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.''\N\NFor the entire collection see [Zbl 1540.58001].