Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces |
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Authors: | S. I. Adyan F. Grunewald J. Mennicke A. L. Talambutsa |
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Affiliation: | 1. Steklov Mathematics Institute, Russian Academy of Science, Russia 2. Mathematishes Institut der Heinrich-Heine-Universit?t, Düsseldorf, Russia 3. Fakult?t für Mathematik der Universit?t Bielefeld, Russia
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Abstract: | 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 ?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, ?) consisting of matrices that contain only 0 and 1. |
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