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

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(Σ_g) 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(Σ_g)). 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(Σ_g))/Inn(π₁(Σ_g)). 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.