Generating the spin mapping class group by Dehn twists

Hamenstädt, Ursula

Annales Henri Lebesgue, Tome 4 (2021), pp. 1619-1658.

Soit $\varphi$ une $\mathbb{Z}/2\mathbb{Z}$-structure de spin sur une surface fermée orientée ${\Sigma }_g$ de genre $g\ge 4$. Nous déterminons une partie génératice du stabilisateur de $\varphi$ dans le groupe de classes d’homéomorphismes de ${\Sigma }_g$, composée de twists de Dehn autour d’un ensemble explicite de $2g+1$ courbes de ${\Sigma }_g$. Lorsque $g=3$, nous déterminons une partie génératrice du stabilisateur d’une $\mathbb{Z}/4\mathbb{Z}$-structure de spin impaire, formée de twist de Dehn autour de $6$ courbes.

Let $\varphi$ be a $\mathbb{Z}/2\mathbb{Z}$-spin structure on a closed oriented surface ${\Sigma }_g$ of genus $g\ge 4$. We determine a generating set of the stabilizer of $\varphi$ in the mapping class group of ${\Sigma }_g$ consisting of Dehn twists about an explicit collection of $2g+1$ curves on ${\Sigma }_g$. If $g=3$ then we determine a generating set of the stabilizer of an odd $\mathbb{Z}/4\mathbb{Z}$-spin structure consisting of Dehn twists about a collection of $6$ curves.

Classification : 30F30, 30F60, 37B10, 37B40
Mots clés : Spin mapping class group, Dehn twists, curve systems, group generators

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