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Algebraic properties of the Teichmüller modular group. (English. Russian original) Zbl 0586.20026
Sov. Math., Dokl. 29, 288-291 (1984); translation from Dokl. Akad. Nauk SSSR 275, 786-789 (1984).
Let \(X_ g\) be a closed orientable surface of genus \(g\). By definition, the Teichmüller modular group \(\text{Mod}_ g\) of genus \(g\) is the group of diffeomorphisms \(X_ g\to X_ g\) considered up to isotopy.
In the note various informations about the algebraic properties of \(\text{Mod}_ g\) are obtained. In particular an analogue of the Tits theorem about free subgroups of linear groups is obtained; it is announced that when \(g\geq 2\) any automorphism of \(\text{Mod}_ g\) is inner and the group \(\text{Mod}_ g\) is not isomorphic to any arithmetic group. The author obtains also the first nontrivial estimates of the virtual cohomological dimension of \(\text{Mod}_ g\). The proofs are based on a study of the action of \(\text{Mod}_ g\) on two different boundaries of genus \(g\): the Thurston boundary and the Harvey boundary.
Detailed proofs have been published [in Leningr. Otd. Mat. Inst. Steklova preprint E-1-85 ”Algebraic properties of the mapping class groups of surfaces”].
Reviewer: G.A.Margulis

20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57R50 Differential topological aspects of diffeomorphisms
20F38 Other groups related to topology or analysis
20J05 Homological methods in group theory
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)