Unsmoothable group actions on compact one-manifolds

Abstract

We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by $C^{1+\mathrm{bv}}$ 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 $C^{1+\mathrm{bv}}$ 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$(F_n)$ or Out$(F_n)$ for $n\ge 3$, 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 $C^{1+\mathrm{bv}}$ diffeomorphisms on a compact one-manifold.