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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Notes
Some of the results of this work strongly complements this. For instance, as we state below, the semiconjugacy is necessarily a (topological) conjugacy, which means that the semiconjugating map is actually a homeomorphism.
Recall that a group is said to be locally indicable if every nontrivial, finitely-generated subgroup has a surjective homomorphism onto \(\mathbb Z\). Every such group admits a faithful action by homeomorphisms of the interval provided it is countable; see [29].
In general, the conjugacy above is not smooth at the endpoints even in the real-analytic case: see [7] for a very complete discussion on this.
This is well-known and follows from the unique ergodicity of a together with that the mean of \(\log (Da)\) with resepect to the unique invariant probability measure equals zero; see [20, Proposition I.I, Chapitre VI].
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Acknowledgments
We thank L. Arenas and A. Zeghib for useful discussions related to Sects. 5 and 1.3, S. Matsumoto and A. Wilkinson for their interest on this work, and the anonymous referee for pointing out to us an error in the original version of this work as well as several points to improve. All the authors were funded by the Center of Dynamical Systems and Related Fields (Anillo Project 1103, CONICYT), and would also like to thank UCN for the hospitality during the VIII Dynamical Systems School held at San Pedro de Atacama (July 2013), where this work started taking its final form. C. Bonatti would like to thank Chicago University for its hospitality during the stay which started his interest on this subject. I. Monteverde would like to thank Univ. of Santiago for the hospitality during his stay in July 2013, and acknowledges the support of PEDECIBA Matemática, Uruguay. A. Navas would like to thank Univ. of Bourgogne for the hospitality during different stages of this work, and acknowledges the support of the FONDECYT Project 1120131. C. Rivas acknowledges the support of the CONICYT Inserción Project 79130017.
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Bonatti, C., Monteverde, I., Navas, A. et al. Rigidity for \(C^1\) actions on the interval arising from hyperbolicity I: solvable groups. Math. Z. 286, 919–949 (2017). https://doi.org/10.1007/s00209-016-1790-y
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DOI: https://doi.org/10.1007/s00209-016-1790-y