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

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, ℤ) consisting of matrices that contain only 0 and 1.