Deformation Space Analogies between Kleinian Reflection Groups and Rational Maps

We will describe an explicit correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps acting on the Riemann sphere. This correspondence has several dynamical and parameter space consequences. To illustrate these, we will discuss some striking similarities between the deformation spaces of these two classes of conformal dynamical systems, including an analogue of Thurston’s compactness theorem for anti-holomorphic rational maps and relations between the global topology of the corresponding deformation spaces. Based on joint work with Russell Lodge and Yusheng Luo.

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