Let be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least . Then any action of the mapping class group of on the circle is trivial.
The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have faithful actions on the circle. We also prove that for , any action of or on the circle factors through an action of .
"$C^1$ actions on the mapping class groups on the circle." Algebr. Geom. Topol. 8 (2) 935 - 944, 2008. https://doi.org/10.2140/agt.2008.8.935