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

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 $g\ge 2$, 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 $g\ge 3$, all finite actions are trivial. This answers a question of Farb.