Le groupe d'automorphismes du groupe modulaire
Annales de l'Institut Fourier, Volume 37 (1987) no. 2, p. 19-31

We give a new and simple proof of the fact that the full mapping class group M g * of a closed oriented surface of genus g3 is complete (this is known as Ivanov’s theorem). In studying the action of an automorphism on finite subgroups of M g * , one remarks that hyperelliptic involutions are mapped onto themselves. The argument also uses Dyer and Grossmann’s result asserting that the outer group of automorphism of the braid group B n is isomorphic to Z 2 . The proof extends to the case of surfaces with one puncture. Unfortunately, our method doesn’t prove the theorem of Ivanov in the case of surfaces with finitely many punctures.

Le but de cet article est de donner une autre démonstration plus simple du théorème d’Ivanov (Théorème 1) qui assure que le groupe M g * de toutes les difféotopies d’une surface F g orientable et fermée de genre g2 est complet. En étudiant l’action d’un automorphisme quelconque du groupe M g * sur les difféotopies d’ordre fini, on montre que les involutions hyperelliptiques sont globalement préservées. Le théorème d’Ivanov est alors une conséquence d’un résultat de Dyer et Grossmann qui affirm que le groupe des automorphismes extérieurs du groupe des tresses est isomorphe à Z 2 . La démonstration s’adapte aussi au cas des surfaces pointées.

@article{AIF_1987__37_2_19_0,
     author = {Tchangang, Tambekou Roger},
     title = {Le groupe d'automorphismes du groupe modulaire},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {37},
     number = {2},
     year = {1987},
     pages = {19-31},
     doi = {10.5802/aif.1084},
     zbl = {0584.57019},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1987__37_2_19_0}
}
Le groupe d'automorphismes du groupe modulaire. Annales de l'Institut Fourier, Volume 37 (1987) no. 2, pp. 19-31. doi : 10.5802/aif.1084. http://www.numdam.org/item/AIF_1987__37_2_19_0/

[1] E. Artin, Braids and Permutations, Annals of Math., 48 (1947), 643-649. | MR 9,6c | Zbl 0030.17802

[2] J. Birman and H. Hilden, On the mapping class group of closed surfaces as covering spaces, Annals of Math. Studies, 66 (1971), 81-115. | MR 45 #1169 | Zbl 0217.48602

[3] J. Dyer and E. Grossman, Automorphisms groups of braids groups, Amer. J. of Math., 103 (1981), 1151-1169. | MR 82m:20041 | Zbl 0476.20026

[4] H. M. Farkas, Automorphisms of compact Riemann surfaces, Annals of Math. Studies, 97 (1974), 121-144. | MR 51 #8402 | Zbl 0293.30019

[5] A. Fathi, F. Laudenbach et V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque, 66-67 (1979). | Numdam | MR 82m:57003 | Zbl 0406.00016

[6] W.J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math., 17 (1966), 86-97. | MR 34 #1511 | Zbl 0156.08901

[7] S. P. Humphries, Generators for the mapping class group, Topology of low dimensional manifolds, Lecture Notes in Math., 722 (1979), 44-47. | MR 80i:57010 | Zbl 0732.57004

[8] N. V. Ivanov, Algebraic properties of the Teichmüller modular group, Soviet Math., Dokl., 29 (1984), 288-291. | Zbl 0586.20026

[9] D. Johnson, The structure of the Torelli group I : a finite set of generators for I. Preprint (1980).

[10] J. Mccarthy, Automorphisms of surface mapping class groups. Manuscrit (1984). | Zbl 0594.57007

[11] J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc., 68 (1978), 347-350. | MR 58 #13045 | Zbl 0391.57009

[12] T. R. Tchangang, Groupe de Torelli et Stratification de l'espace de Teichmüller d'une surface de genre deux. Thèse, Strasbourg, 1985.

[13] H. Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Math., 875 (1981), V. | MR 86g:57001 | Zbl 0472.57006