#### Volume 24, issue 3 (2020)

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

### Kathryn Mann and Maxime Wolff

Geometry & Topology 24 (2020) 1211–1223
##### Abstract

The mapping class group ${Mod}_{g,1}$ of a surface with one marked point can be identified with an index two subgroup of $Aut\left({\pi }_{1}{\Sigma }_{g}\right)$. 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 ${\pi }_{1}{\Sigma }_{g}$, or factors through a finite cyclic group. For $g\ge 3$, all finite actions are trivial. This answers a question of Farb.

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##### Keywords
mapping class group, Gromov boundary, surface group, rigidity, homeomorphisms of the circle, Euler class
##### Mathematical Subject Classification 2010
Primary: 57M60
Secondary: 20F34, 57M07