Geometry & Topology

Hyperbolicity as an obstruction to smoothability for one-dimensional actions

Christian Bonatti, Yash Lodha, and Michele Triestino

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Abstract

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 xpqx, 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.

Article information

Source
Geom. Topol., Volume 23, Number 4 (2019), 1841-1876.

Dates
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
https://projecteuclid.org/euclid.gt/1563242521

Digital Object Identifier
doi:10.2140/gt.2019.23.1841

Mathematical Reviews number (MathSciNet)
MR3988090

Zentralblatt MATH identifier
07094909

Subjects
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

Keywords
group actions on the interval piecewise-projective homeomorphisms hyperbolic dynamics

Citation

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


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