On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps

Abstract

In a previous paper [LLM20], we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston's compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces. Contents 1. Introduction 1 2. Degeneration of anti-Blaschke products 8 3. Realization of (d + 1)-ended ribbon trees 19 4. Boundedness and mutual interaction of deformation spaces 29 5. Markov partitions and monodromy representations 39 Appendix A. Laminations, automorphisms and accesses 42 Appendix B. Shared matings, self-bumps and disconnected roots 47 References 52