Subgroups of Teichmüller modular groups and their Frattini subgroups
type: Item
instance of: scholarly article
title: Subgroups of Teichmüller modular groups and their Frattini subgroups
zbMATH Open document ID: 0629.57006
DOI: 10.1007/BF01078033
publication date: 1987-01-01T00:00:00Z
review text: The paper deals with subgroups of the modular group \(Mod_ S\) of a compact Riemann surface S. Its method is based on a study of the action of \(Mod_ S\) on Thurston's boundary of the Teichmüller space \(T(S)\). So, for the irreducible case, the author points out: Theorem 1. If \(G\subset Mod_ S\) is an infinite irreducible subgroup then G contains a pseudo-Anosov element. Theorem 2. If \(G\subset Mod_ S\) is an infinite irreducible acyclic subgroup then it contains two pseudo-Anosov elements, f and g, such that \(fix(g)\cap fix(f)=\emptyset\). Similar results hold in the reducible case, too. As applications of these results the author obtains new proofs of some of his older results and of the following analogy of Platonov's theorem: Theorem 4. If \(G\subset Mod_ S\) is a finitely generated subgroup then its Frattini subgroup \(F(G)\) (= intersection of all maximal subgroups of G) is nilpotent and almost abelian. [See also \textit{D. D. Long}, Math. Proc. Camb. Philos. Soc. 99, 79-87 (1986; Zbl 0584.57008); and \textit{J. McCarthy}, Trans. Am. Math. Soc. 291, 583-612 (1985; Zbl 0579.57006).]
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20F38
zbMATH DE Number: 4024306
zbMATH Keywords: subgroups of the modular group of a compact Riemann surface
zbMATH Keywords: Thurston's boundary of the Teichmüller space
zbMATH Keywords: pseudo-Anosov element
zbMATH Keywords: Frattini subgroup
author: Nikolai V. Ivanov
reviewed by: Boris N. Apanasov
MaRDI profile type: Publication
cites work: Structures of Elliptic Irreducible Subgroups of the Modular Group
cites work: A note on the normal subgroups of mapping class groups
full work available at URL: https://doi.org/10.1007/bf01078033
OpenAlex ID: W2046259039
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4024306