We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by diffeomorphisms on the circle, which generalizes a result of Farb–Franks, and which parallels a result of Ghys and Burger–Monod concerning differentiable actions of higher rank lattices on the circle. This answers a question of Farb, which has its roots in the work of Nielsen. We prove this result by showing that if a right-angled Artin group acts faithfully by diffeomorphisms on a compact one-manifold, then its defining graph has no subpath of length 3. As a corollary, we also show that no finite index subgroup of Aut or Out for , of the Torelli group for genus at least 3, and of each term of the Johnson filtration for genus at least 5, can act faithfully by diffeomorphisms on a compact one-manifold.
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Hyungryul Baik, Sang-hyun Kim, Thomas Koberda, Unsmoothable group actions on compact one-manifolds. J. Eur. Math. Soc. 21 (2019), no. 8, pp. 2333–2353DOI 10.4171/JEMS/886