A Combination Theorem for Covering Correspondences and an Application to Mating Polynomial Maps with Kleinian Groups
نویسنده
چکیده
The simplest version of the Maskit-Klein combination theorems concerns the action of a free product of two finite subgroups of PSL(2,C) on the Riemann sphere Ĉ, when these subgroups have fundamental domains whose interiors together cover Ĉ. We prove an analogous combination theorem for covering correspondences of rational maps, making use of Douady and Hubbard’s Straightening Theorem for polynomial-like maps to describe the structure of the limit sets. We apply our theorem to construct holomorphic correspondences which are matings of polynomial maps with Hecke groups Cp ∗ Cq , and we show how it may also be applied to the analysis of separable correspondences. 1. Covering correspondences and transversals for rational maps For any rational map Q : Ĉ→ Ĉ of degree q we define • Cov to be the (q : q) correspondence (multi-valued map) z → w where w runs through all values such that
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