Rigidity for $$C^1$$C1 actions on the interval arising from hyperbolicity I: solvable groups

By C. Bonatti, I. Monteverde, Andrés Navas, Cristóbal Rivas. 2013

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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 $$C^1$$C1 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 $$C^1$$C1 diffeomorphisms of the interval.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a4edc8b18c1315aa9f1312e028a3eb5fc9babe72