an:03939588
Zbl 0586.20026
Ivanov, N. V.
Algebraic properties of the Teichmüller modular group
EN
Sov. Math., Dokl. 29, 288-291 (1984); translation from Dokl. Akad. Nauk SSSR 275, 786-789 (1984).
0197-6788
1984
j
20H05 32G15 57R50 20F38 20J05 30F35
closed orientable surface; Teichmüller modular group; group of diffeomorphisms; Tits theorem; automorphism; virtual cohomological dimension; action; Thurston boundary; Harvey boundary
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''].
G.A.Margulis