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

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.