Geometry & Topology
- Geom. Topol.
- Volume 23, Number 4 (2019), 1841-1876.
Hyperbolicity as an obstruction to smoothability for one-dimensional actions
Ghys and Sergiescu proved in the 1980s that Thompson’s group , and hence , admits actions by diffeomorphisms of the circle. They proved that the standard actions of these groups are topologically conjugate to a group of 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 diffeomorphisms.
Furthermore, we show that the group of Lodha and Moore has no nonabelian action on the interval. We also show that many of Monod’s groups , for instance when is such that contains a rational homothety , do not admit a action on the interval. The obstruction comes from the existence of hyperbolic fixed points for actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable.
Geom. Topol., Volume 23, Number 4 (2019), 1841-1876.
Received: 18 June 2017
Revised: 17 July 2018
Accepted: 24 November 2018
First available in Project Euclid: 16 July 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 57M60: Group actions in low dimensions
Secondary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 43A07: Means on groups, semigroups, etc.; amenable groups
Bonatti, Christian; Lodha, Yash; Triestino, Michele. Hyperbolicity as an obstruction to smoothability for one-dimensional actions. Geom. Topol. 23 (2019), no. 4, 1841--1876. doi:10.2140/gt.2019.23.1841. https://projecteuclid.org/euclid.gt/1563242521