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

TL;DR

This paper explores the similarities between deformation spaces of Kleinian reflection groups and antiholomorphic rational maps, establishing analogues of Thurston's theorem and analyzing their monodromy representations.

Contribution

It introduces a new analogy between deformation spaces of Kleinian groups and antiholomorphic rational maps, including a Thurston-type compactness theorem.

Findings

Deformation spaces of the two classes share many similarities.

An analogue of Thurston's compactness theorem is established.

Interaction and monodromy of deformation spaces are characterized.

Abstract

In a previous paper, 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.