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Title:
Matings, Correspondences, and Schwarz Reflections
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Abstract:
We will introduce two notions of matings of groups and polynomials. The first framework, which can be seen as an analogue of the Douady-Hubbard theory of polynomial mating, allows one to conformally mate anti-holomorphic polynomials with reflection groups, and realize them as Schwarz reflection maps in quadrature domains. In the second set-up, we will formulate a general theory of Bullett-Penrose correspondences in the anti-holomorphic world by producing correspondences on the Riemann sphere that are matings of abstract Hecke groups with anti-holomorphic polynomials. The exposition of the mating phenomena will be accompanied by existence theorems and concrete examples.
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