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Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces. (English. Russian original) Zbl 1128.20021

Math. Notes 81, No. 2, 147-155 (2007); translation from Mat. Zametki 81, No. 2, 163-173 (2006).
The mapping class group of a closed orientable surface \(\Sigma_g\) of genus \(g\) is isomorphic to the outer automorphism group \(\operatorname{Aut}^+(\pi_1(\Sigma_g))/\text{Inn}(\pi_1(\Sigma_g))\) of its fundamental group. Let \(N\) be the stabilizer of the commutator product \(w=[s_1,t_1]\cdots[s_g,t_g]\) in the free group \(F_{2g}\) on \(2g\) generators (so \(\pi_1(\Sigma_g)\) is obtained from \(F_{2g}\) by adding the single relation \(w=1\)).
In the present paper, a presentation in terms of explicit generators and relations of the kernel of the natural map (which is known to be an epimorphism) from \(N\) to \(\operatorname{Aut}^+(\pi_1(\Sigma_g))/\text{Inn}(\pi_1(\Sigma_g))\) is obtained (and this turns out to be a standard presentation of the fundamental group of a 3-dimensional Seifert fibered manifold without exceptional fibers). Related results are contained in a paper of H. Zieschang [in Lond. Math. Soc. Lect. Note Ser. 95, 206-213 (1985; Zbl 0582.57006)] where it is shown that the kernel of the natural epimorphism from \(N\) to \(\operatorname{Aut}^+(\pi_1(\Sigma_g))\) is cyclic, generated by the inner automorphism with \(w\).
In addition, a subgroup of the stabilizer \(N\) is found which is isomorphic to the braid group \(B_g\) on \(g\) strings, and which maps under Abelianization of the free group \(F_{2g}\) onto the subgroup of matrices in the Weyl group of the symplectic group \(\text{Sp}(2g,\mathbb{Z})\) having only entries 0 and 1.

MSC:

20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
20F28 Automorphism groups of groups
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 0582.57006
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References:

[1] M. Dehn, ”Die Gruppe der Abbildungsklassen,” Acta Math. 69, 135–206 (1938). · Zbl 0019.25301
[2] 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). · Zbl 0131.20801
[3] 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.
[4] A. Hatcher and W. Thurston, ”A presentation for the mapping class group of a closed orientable surface,” Topology 19(3), 221–237 (1980). · Zbl 0447.57005
[5] B. Wajnryb, ”An elementary approach to the mapping class group of a surface,” Geom. Topol. 3, 405–466 (1999). · Zbl 0947.57015
[6] H. Zieschang, E. Vogt, and H.-D. Coldewey, Surfaces and Planar Discontinuities (Springer-Verlag, Berlin, 1980; Nauka, Moscow, 1988).
[7] W. Magnus, A. Karras, and D. Solitar, Combinatorial Group Theory (Interscience Publishers, 1966; Nauka, Moscow, 1974).
[8] 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.
[9] H. Seifert, ”Topologie dreidimensionaler gefaserter Räume,” Acta Math. 60, 147–238 (1933). · Zbl 0006.08304
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