Automorphisms of Free Groups and the Mapping Class Groups of Closed Compact Orientable Surfaces

Let $N$ be the stabilizer of the word $w=s_1t_1s_1^{-1}t_1^{-1}\dots s_gt_gs_g^{-1}t_g^{-1}$ in the group of automorphisms $\operatorname{Aut}(F_{2g})$ of the free group with generators $\{s_i,t_i\}_{i=1,\dots,g}$. 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 $S_i,T_i\in N$ determining the conjugation by the generators $s_i$, $t_i$ in $\operatorname{Aut}(\pi_1(\Sigma_g))$. Along with an element $\beta\in N$, realizing the conjugation by $w$, they generate the kernel of the natural epimorphism of the group $N$ on the mapping class group $M_{g,0}=\operatorname{Aut}(\pi_1(\Sigma_g))/\operatorname{Inn}(\pi_1(\Sigma_g))$. We find the system of defining relations for this kernel in the generators $S_1$, …, $S_g$, $T_1$, …, $T_g$, $\alpha$. 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 $\operatorname{Sp}(2g,\mathbb{Z})$ consisting of matrices that contain only $0$ and $1$.