Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces

Adyan SI, Grunewald F, Mennicke J, Talambutsa AL (2007)
Mathematical Notes 81(1-2): 147-155.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Adyan, Sergei Iwanowitsch; Grunewald, Fritz; Mennicke, JensUniBi; Talambutsa, Alexey Leonidovich
Abstract / Bemerkung
Let N be the stabilizer of the word w = s(1)t(1)s(1)(-1)t(1)(-1) ...s(g)t(g)s(g)(-1)t(g)(-1) in the group of automorphisms Aut(F-2g) of the free group with generators {s(i), t(i)}(i=1),...,g. The fundamental group pi(1)(Sigma(g)) of a two-dimensional compact orientable closed surface of genus g in generators {s(i),t(i)} is determined by the relation w = 1. In the present paper, we find elements S-i, T-i is an element of N determining the conjugation by the generators s(i), t(i) in Aut(pi(1) (Sigma(g))). Along with an element beta is an element of 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,M-0 = Aut(pi(1)(Sigma(g)))/Inn(pi(1)(Sigma(g))). We find the system of defining relations for this kernel in the generators S-1,..., Sg, T-1,..., T-g, alpha. 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.
Stichworte
mapping class group; fundamental; group; automorphism; homeomorphism; generators and defining relations; closed compact orientable surface
Erscheinungsjahr
2007
Zeitschriftentitel
Mathematical Notes
Band
81
Ausgabe
1-2
Seite(n)
147-155
ISSN
0001-4346
Page URI
https://pub.uni-bielefeld.de/record/1595214

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Adyan SI, Grunewald F, Mennicke J, Talambutsa AL. Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces. Mathematical Notes. 2007;81(1-2):147-155.
Adyan, S. I., Grunewald, F., Mennicke, J., & Talambutsa, A. L. (2007). Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces. Mathematical Notes, 81(1-2), 147-155. https://doi.org/10.1134/S0001434607010178
Adyan, S. I., Grunewald, F., Mennicke, J., and Talambutsa, A. L. (2007). Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces. Mathematical Notes 81, 147-155.
Adyan, S.I., et al., 2007. Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces. Mathematical Notes, 81(1-2), p 147-155.
S.I. Adyan, et al., “Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces”, Mathematical Notes, vol. 81, 2007, pp. 147-155.
Adyan, S.I., Grunewald, F., Mennicke, J., Talambutsa, A.L.: Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces. Mathematical Notes. 81, 147-155 (2007).
Adyan, Sergei Iwanowitsch, Grunewald, Fritz, Mennicke, Jens, and Talambutsa, Alexey Leonidovich. “Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces”. Mathematical Notes 81.1-2 (2007): 147-155.

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