Combining Rational maps and Kleinian groups via orbit equivalence

Abstract

We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings has one component corresponding to Bowen-Series maps and one corresponding to higher Bowen-Series maps. We also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers boundary groups that are mateable in our framework.

arXiv commentv 2: 70 pages, 9 figures; final version, accepted in Proceedings of the London Math. Soc. Some examples and expository details from the earlier version have now been moved to arXiv:2302.09827

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Cited by in corpus (3)

• The Sullivan dictionary and Bowen-Series maps
• Matings, holomorphic correspondences, and a Bers slice
• Combination Theorems in Groups, Geometry and Dynamics