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

Javier Ribón ()
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Javier Ribón: Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis s/n

Chapter Chapter 7 in Handbook of Geometry and Topology of Singularities VI: Foliations, 2024, pp 231-265 from Springer

Abstract: Abstract 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 $$G_u$$ of G consisting of its unipotent elements. In such a case we show that if G ∕ G u $$G/G_u$$ and G u $$G_u$$ are finitely generated and G u $$G_u$$ 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.

Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-54172-8_7

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DOI: 10.1007/978-3-031-54172-8_7

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