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 Mat. Zametki, 2007, Volume 81, Issue 2, Pages 163–173 (Mi mz3544)

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

S. I. Adiana, F. Grunevaldb, J. Mennickec, A. L. Talambutsaa

a Steklov Mathematical Institute, Russian Academy of Sciences
b Heinrich-Heine-Universität Düsseldorf
c Bielefeld University

Abstract: Let $N$ be the stabilizer of the word $w=s_1t_1s_1^{-1}t_1^{-1}…s_gt_gs_g^{-1}t_g^{-1}$ in the group of automorphisms $\operatorname{Aut}(F_{2g})$ of the free group with generators $\{s_i,t_i\}_{i=1,…,g}$. 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 $1$.

Keywords: mapping class group, closed compact orientable surface, fundamental group, automorphism, homeomorphism, generators and defining relations

DOI: https://doi.org/10.4213/mzm3544

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English version:
Mathematical Notes, 2007, 81:2, 147–155

Bibliographic databases:

Document Type: Article
UDC: 512.54

Citation: S. I. Adian, F. Grunevald, J. Mennicke, A. L. Talambutsa, “Automorphisms of Free Groups and the Mapping Class Groups of Closed Compact Orientable Surfaces”, Mat. Zametki, 81:2 (2007), 163–173; Math. Notes, 81:2 (2007), 147–155

Citation in format AMSBIB
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