JournalsjemsVol. 21, No. 8pp. 2333–2353

Unsmoothable group actions on compact one-manifolds

  • Hyungryul Baik

    Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea
  • Sang-hyun Kim

    Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea
  • Thomas Koberda

    University of Virginia, Charlottesville, USA
Unsmoothable group actions on compact one-manifolds cover
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Abstract

We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by C1+bvC^{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 C1+bvC^{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(Fn)(F_n) or Out(Fn)(F_n) for n3n \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 C1+bvC^{1+\mathrm {bv}} diffeomorphisms on a compact one-manifold.

Cite this article

Hyungryul Baik, Sang-hyun Kim, Thomas Koberda, Unsmoothable group actions on compact one-manifolds. J. Eur. Math. Soc. 21 (2019), no. 8, pp. 2333–2353

DOI 10.4171/JEMS/886