Unsmoothable group actions on compact one-manifolds

TL;DR

This paper proves that certain complex groups, including mapping class groups and automorphism groups of free groups, cannot act faithfully by smooth diffeomorphisms with bounded variation derivatives on the circle, revealing fundamental limitations on their actions.

Contribution

It establishes new restrictions on smooth actions of complex groups on one-dimensional manifolds, extending previous results and answering longstanding questions in the field.

Findings

Finite index subgroups of complex groups cannot act faithfully by $C^{1+\mathrm{bv}}$ diffeomorphisms on the circle.

Right-angled Artin groups with certain graph properties cannot act faithfully by such diffeomorphisms.

Specific automorphism and Torelli groups are also shown to have no faithful $C^{1+\mathrm{bv}}$ 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 three. As a corollary, we also show that no finite index subgroup of $\textrm{Aut}(F_n)$ and $\textrm{Out}(F_n)$ for $n\geq 3$, 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.