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arXiv · 1309.5277

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

C. Bonatti·I. Monteverde·A. Navas·C. Rivas
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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 conjugated 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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C. Bonatti, I. Monteverde, A. Navas, C. Rivas. 2016-07-19. Rigidity for $C^1$ actions on the interval arising from hyperbolicity I: solvable groups. https://arxiv.org/abs/1309.5277

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