Subgroups of Teichmüller modular groups and their Frattini subgroups
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).]
subgroups of the modular group of a compact Riemann surface
Thurston's boundary of the Teichmüller space
scientific article; zbMATH DE number 4024306