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# Rigidity for $C^1$ actions on the interval arising from hyperbolicity I : solvable groups

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Abstract : We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugate to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result concerning the multipliers of the homotheties, despite the fact that the conjugacy is not necessarily smooth. Some consequences for non-solvable groups are proposed. In particular, we give new proofs/examples yielding the existence of finitely-generated, locally-indicable groups with no faithful action by $C^1$ diffeomorphisms of the interval.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01582508
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Soumis le : mercredi 6 septembre 2017 - 10:08:39
Dernière modification le : mardi 18 janvier 2022 - 14:50:08

### Citation

Christian Bonatti, Ignacio Monteverde, Andrés Navas, Cristobal Rivas. Rigidity for $C^1$ actions on the interval arising from hyperbolicity I : solvable groups. Mathematische Zeitschrift, Springer, 2017, 286 (3-4), pp.919 - 949. ⟨10.1007/s00209-016-1790-y⟩. ⟨hal-01582508⟩

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