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Article Dans Une Revue Mathematische Zeitschrift Année : 2017

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

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Christian Bonatti
Ignacio Monteverde
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Andrés Navas
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Cristobal Rivas
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#### Résumé

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.

#### Domaines

Mathématiques [math]

### Dates et versions

hal-01582508 , version 1 (06-09-2017)

### Identifiants

• HAL Id : hal-01582508 , version 1
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• DOI :

### Citer

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

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