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Subgroups of Teichmüller modular groups and their Frattini subgroups. (English. Russian original) Zbl 0629.57006
Funct. Anal. Appl. 21, No. 1-3, 154-155 (1987); translation from Funkts. Anal. Prilozh. 21, No. 2, 76-77 (1987).
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 D. D. Long, Math. Proc. Camb. Philos. Soc. 99, 79-87 (1986; Zbl 0584.57008); and J. McCarthy, Trans. Am. Math. Soc. 291, 583-612 (1985; Zbl 0579.57006).]
Reviewer: B.N.Apanasov

MSC:
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57R50 Differential topological aspects of diffeomorphisms
20E07 Subgroup theorems; subgroup growth
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20F38 Other groups related to topology or analysis
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[1] A. Fathi, F. Laudenbach, and V. Poenaru, Asterisque, No. 66-67 (1979).
[2] J. Gilman, Proc. London Math. Soc.,47, No. 1, 27-42 (1983). · Zbl 0522.30035
[3] N. V. Ivanov, Dokl. Akad. Nauk SSSR,275, No. 4, 786-789 (1984).
[4] N. V. Ivanov, ”Algebraic properties of the mapping class groups of surfaces,” LOMI Preprints, E-1-85, Leningrad (1985).
[5] D. D. Long, Math. Proc. Camb. Philos. Soc.,99, No. 1, 79-87 (1986). · Zbl 0584.57008
[6] J. McCarthy, Trans. Am. Math. Soc.,291, No. 2, 583-612 (1985).
[7] V. P. Platonov, Dokl. Akad. Nauk SSSR,171, No. 4, 798-801 (1966).
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