Hyperbolicity as an obstruction to smoothability for one-dimensional actions

Abstract : Ghys and Sergiescu proved in the $80$s that Thompson's groups $F$ and $T$ admit actions by $C^{\infty}$ diffeomorphisms of the interval. 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-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-Sergiescu, we prove that the groups of Monod and Lodha-Moore are not topologically conjugate to a group of $C^1$ diffeomorphisms. Furthermore, we show that the group of Lodha-Moore has no nonabelian $C^1$ action on the interval. We also show that Monod's groups $H(A)$, in the case where $\mathsf{PSL}(2,A)$ contains a rational homothety $x\mapsto \tfrac{p}{q}x$, 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 are not smoothable.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01542608
Contributeur : Michele Triestino <>
Soumis le : lundi 17 juillet 2017 - 02:27:59
Dernière modification le : lundi 24 juillet 2017 - 14:35:02

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Distributed under a Creative Commons Paternité 4.0 International License

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  • HAL Id : hal-01542608, version 2
  • ARXIV : 1706.05704

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Christian Bonatti, Yash Lodha, Michele Triestino. Hyperbolicity as an obstruction to smoothability for one-dimensional actions. 2017. 〈hal-01542608v2〉

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