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

TL;DR

This paper proves a rigidity result for actions of the mapping class group on the circle, showing they are either essentially the natural boundary action or trivial, especially for genus at least 3.

Contribution

It establishes a classification of circle actions of the mapping class group, confirming that all finite actions are trivial for genus at least 3, and answers a question posed by Farb.

Findings

Any action of $ ext{Mod}_{g,1}$ on $S^1$ is either semi-conjugate to the natural boundary action or factors through a finite cyclic group.

For genus $g ext{ } ext{≥} ext{ } 3$, all finite actions are trivial.

The result confirms a conjecture about the rigidity of these group actions.

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.