Rigidity of mapping class group actions on $S^1$

Kathryn Mann; Maxime Wolff

doi:10.2140/gt.2020.24.1211

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arXiv · 1808.02979

Rigidity of mapping class group actions on $S^1$

Kathryn Mann·Maxime Wolff

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Abstract

The mapping class group $\mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $\mathrm{Aut}(\pi_1 \Sigma_g)$. For a surface of genus $g \geq 2$, we show that any action of $\mathrm{Mod}_{g, 1}$ on the circle is either semi-conjugate to its natural action on the Gromov boundary of $\pi_1 \Sigma_g$, or factors through a finite cyclic group. For $g \geq 3$, all finite actions are trivial. This answers a question of Farb.

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Kathryn Mann, Maxime Wolff. 2018-08-09. Rigidity of mapping class group actions on $S^1$. https://doi.org/10.2140/gt.2020.24.1211

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