Abstract
Let N be the stabilizer of the word w = s 1 t 1 s −11 t −11 … s g t g s −1g t −1g in the group of automorphisms Aut(F 2g ) of the free group with generators ⨑ub;s i, t i⫂ub; i=1,…,g . The fundamental group π1(Σg) of a two-dimensional compact orientable closed surface of genus g in generators ⨑ub;s i, t i⫂ub; is determined by the relation w = 1. In the present paper, we find elements S i, T i ∈ N determining the conjugation by the generators s i, t i in Aut(π1(Σg)). Along with an element β ∈ N, realizing the conjugation by w, they generate the kernel of the natural epimorphism of the group N on the mapping class group M g,0 = Aut(π1(Σg))/Inn(π1(Σg)). We find the system of defining relations for this kernel in the generators S 1, …, S g, T 1, …, T g, α. In addition, we have found a subgroup in N isomorphic to the braid group B g on g strings, which, under the abelianizing of the free group F 2g , is mapped onto the subgroup of the Weyl group for Sp(2g, ℤ) consisting of matrices that contain only 0 and 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Dehn, “Die Gruppe der Abbildungsklassen,” Acta Math. 69, 135–206 (1938).
W. B. R. Lickorish, “A finite set of generators for the homeotopy group of a 2-manifold,” Proc. Cambridge Philos. Soc. 60, 769–778 (1964); “Corrigendum: On the homeotopy group of a 2-manifold,” Proc. Cambridge Philos. Soc. 62, 679–681 (1966).
S. P. Humphries, “Generators for the mapping class group,” in Proceedings of Second Sussex Conf on Topology of Low-Dimensional Manifolds, Chelwood Gate, 1977, Lecture Notes in Math. (Springer, Berlin, 1979), Vol. 722, pp. 44–47.
A. Hatcher and W. Thurston, “A presentation for the mapping class group of a closed orientable surface,” Topology 19(3), 221–237 (1980).
B. Wajnryb, “An elementary approach to the mapping class group of a surface,” Geom. Topol. 3, 405–466 (1999).
H. Zieschang, E. Vogt, and H.-D. Coldewey, Surfaces and Planar Discontinuities (Springer-Verlag, Berlin, 1980; Nauka, Moscow, 1988).
W. Magnus, A. Karras, and D. Solitar, Combinatorial Group Theory (Interscience Publishers, 1966; Nauka, Moscow, 1974).
H. Zieschang, “A note on the mapping class groups of surfaces and planar discontinuous groups,” in Proceedings on Low-Dimensional Topology, Chelwood Gate, 1982, London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press,, Cambridge, 1985), Vol. 95, pp. 206–213.
H. Seifert, “Topologie dreidimensionaler gefaserter Räume,” Acta Math. 60, 147–238 (1933).
Author information
Authors and Affiliations
Additional information
Original Russian Text © S. I. Adyan, F. Grunewald, J. Mennicke, A. L. Talambutsa, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 2, pp. 163–173.
Rights and permissions
About this article
Cite this article
Adyan, S.I., Grunewald, F., Mennicke, J. et al. Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces. Math Notes 81, 147–155 (2007). https://doi.org/10.1134/S0001434607010178
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0001434607010178