Journal of the European Mathematical Society


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Volume 21, Issue 8, 2019, pp. 2333–2353
DOI: 10.4171/JEMS/886

Published online: 2019-04-15

Unsmoothable group actions on compact one-manifolds

Hyungryul Baik[1], Sang-hyun Kim[2] and Thomas Koberda[3]

(1) Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea
(2) Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea
(3) University of Virginia, Charlottesville, USA

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 $n \geq 3$, 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.

Keywords: Mapping class groups, right-angled Artin groups, circle actions

Baik Hyungryul, Kim Sang-hyun, Koberda Thomas: Unsmoothable group actions on compact one-manifolds. J. Eur. Math. Soc. 21 (2019), 2333-2353. doi: 10.4171/JEMS/886