Hyperbolicity as an obstruction to smoothability for one-dimensional actions

Bonatti, Christian; Lodha, Yash; Triestino, Michele

doi:10.2140/gt.2019.23.1841

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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

DOI: 10.2140/gt.2019.23.1841

Abstract

Ghys and Sergiescu proved in the 1980s that Thompson’s group $T$ , and hence $F$ , admits actions by $C^{\infty}$ diffeomorphisms of the circle. They proved that the standard actions of these groups are topologically conjugate to a group of $C^{\infty}$ 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 $C^{1}$ diffeomorphisms.

Furthermore, we show that the group of Lodha and Moore has no nonabelian $C^{1}$ 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 $x \mapsto \frac{p}{q} x$ , do not admit a $C^{1}$ action on the interval. The obstruction comes from the existence of hyperbolic fixed points for $C^{1}$ actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable.

Keywords

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

Mathematical Subject Classification 2010

Primary: 37C85, 57M60

Secondary: 37D40, 37E05, 43A07

References

Publication

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

Authors

Christian Bonatti
CNRS Institut de Mathématiques de Bourgogne (CNRS UMR 5584) Université de Bourgogne Dijon France

Yash Lodha
Section de Mathematiques EPFL Lausanne Switzerland

Michele Triestino
Institut de Mathématiques de Bourgogne (CNRS UMR 5584) Université de Bourgogne Dijon France

Volume 23, issue 4 (2019)