On deformation space analogies between Kleinian reflection groups and antiholomorphic rational maps

Abstract

In a previous paper (Lodge et al. in Forum Math Sigma 10:e3, 2022), 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.