Hyperbolicity as an obstruction to smoothability for one-dimensional actions

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 x↦pqx, 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.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-02194613
Contributeur : Imb - Université de Bourgogne <>
Soumis le : jeudi 25 juillet 2019 - 16:55:14
Dernière modification le : vendredi 26 juillet 2019 - 01:27:48

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Christian Bonatti, Yash Lodha, Michele Triestino. Hyperbolicity as an obstruction to smoothability for one-dimensional actions. Geometry & Topology Monographs, Geometry & Topology Publications, 2019, 23 (4), pp.1841-1876. ⟨https://msp.org/gt/2019/23-4/p04.xhtml⟩. ⟨10.2140/gt.2019.23.1841⟩. ⟨hal-02194613⟩

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