Description of the Zariski-closure of a group of formal diffeomorphisms
type: Item
instance of: scholarly article
title: Description of the Zariski-closure of a group of formal diffeomorphisms
DOI: 10.1007/978-3-031-54172-8_7
author: Javier Ribón
publication date: 2024-09-13T00:00:00Z
review text: 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].
reviewed by: Vagn Lundsgaard Hansen
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 14Bxx
zbMATH DE Number: 7913002
zbMATH Keywords: subgroup of germs of biholomorphisms
zbMATH Keywords: subgroup of formal diffeomorphisms
zbMATH Keywords: Zariski-closure of subgroup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7913002