Combining Rational maps and Kleinian groups via orbit equivalence

TL;DR

This paper introduces a new orbit equivalence framework for mating complex polynomial dynamics with Kleinian surface groups, revealing unique classes of maps and the structure of their Teichmüller spaces.

Contribution

It develops a novel orbit equivalence approach for holomorphic mating, identifies higher Bowen-Series maps, and classifies mateable Kleinian Bers boundary groups.

Findings

Only punctured sphere groups can be mated with polynomials.

Existence of higher Bowen-Series maps expands the class of orbit equivalent maps.

Topological orbit equivalence rigidity fails for Fuchsian groups on the circle.

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 Teichm\"uller 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.