PUB - Publikationen an der Universität Bielefeld

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.