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- Title
Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces.
- Authors
Adyan, S.; Grunewald, F.; Mennicke, J.; Talambutsa, A.
- Abstract
Let N be the stabilizer of the word w = s 1 t 1 s t ... s g t g s t in the group of automorphisms Aut( F 2 g ) 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 2 g , is mapped onto the subgroup of the Weyl group for Sp(2 g, ℤ) consisting of matrices that contain only 0 and 1.
- Subjects
FREE groups; CLASS groups (Mathematics); AUTOMORPHISMS; HOMEOMORPHISMS; MATHEMATICAL mappings; MATHEMATICAL research
- Publication
Mathematical Notes, 2007, Vol 81, Issue 1/2, p147
- ISSN
0001-4346
- Publication type
Article
- DOI
10.1134/S0001434607010178