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Mating quadratic maps with Kleinian groups via quasiconformal surgery. (English) Zbl 1027.37024

Summary: Let \(q:\hat{\mathbb C} \to \hat{\mathbb C}\) be any quadratic polynomial and \(r:C_2*C_3 \to \text{PSL}(2,{\mathbb C})\) be any faithful discrete representation of the free product of finite cyclic groups \(C_2\) and \(C_3\) (of orders \(2\) and \(3\)) having a connected regular set. We show how the actions of \(q\) and \(r\) can be combined, using quasiconformal surgery, to construct a \(2:2\) holomorphic correspondence \(z \to w\), defined by an algebraic relation \(p(z,w)=0\).

MSC:

37F05 Dynamical systems involving relations and correspondences in one complex variable
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
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References:

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