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

Abstract

The mapping class group ${Mod}_{g,1}$ of a surface with one marked point can be identified with an index two subgroup of $Aut\left({\pi }_1{\mathrm{\Sigma }}_g\right)$. For a surface of genus $g\ge 2$, we show that any action of ${Mod}_{g,1}$ on the circle is either semiconjugate to its natural faithful action on the Gromov boundary of ${\pi }_1{\mathrm{\Sigma }}_g$, or factors through a finite cyclic group. For $g\ge 3$, all finite actions are trivial. This answers a question of Farb.