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Hyperbolicity as an obstruction to smoothability for one-dimensional actions

Christian Bonatti, Yash Lodha and Michele Triestino

Geometry & Topology 23 (2019) 1841–1876

Ghys and Sergiescu proved in the 1980s that Thompson’s group T, and hence F, admits actions by C diffeomorphisms of the circle. They proved that the standard actions of these groups are topologically conjugate to a group of C diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha and Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys and Sergiescu, we prove that the groups of Monod and Lodha and Moore are not topologically conjugate to a group of C1 diffeomorphisms.

Furthermore, we show that the group of Lodha and Moore has no nonabelian C1 action on the interval. We also show that many of Monod’s groups H(A), for instance when A is such that PSL(2,A) contains a rational homothety xp qx, do not admit a C1 action on the interval. The obstruction comes from the existence of hyperbolic fixed points for C1 actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable.

group actions on the interval, piecewise-projective homeomorphisms, hyperbolic dynamics
Mathematical Subject Classification 2010
Primary: 37C85, 57M60
Secondary: 37D40, 37E05, 43A07
Received: 18 June 2017
Revised: 17 July 2018
Accepted: 24 November 2018
Published: 17 June 2019
Proposed: Jean-Pierre Otal
Seconded: Leonid Polterovich, Yasha Eliashberg
Christian Bonatti
Institut de Mathématiques de Bourgogne (CNRS UMR 5584)
Université de Bourgogne
Yash Lodha
Section de Mathematiques
Michele Triestino
Institut de Mathématiques de Bourgogne (CNRS UMR 5584)
Université de Bourgogne