Mating Kleinian groups isomorphic to with quadratic polynomials
Conform. Geom. Dyn. 7 (2003), 11-33
Primary 37F45, 37F30, 37F05; Secondary 37F10
May 27, 2003
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Abstract: Given a quadratic polynomial and a representation of in satisfying certain conditions, we will construct a holomorphic correspondence on the sphere (given by a polynomial relation ) that mates the two actions: The sphere will be partitioned into two completely invariant sets and . The set consists of the disjoint union of two sets, and , each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial . This filled Julia set contains infinitely many copies of the filled Julia set of . Suitable restrictions of the correspondence are conformally conjugate to on each of and . The set will not be connected, but it can be joined up using a family of completely invariant curves. The action of the correspondence on the complement of will then be conformally conjugate to the action of on a simply connected subset of its regular set.
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School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK
Received by editor(s):
November 23, 2001
Received by editor(s) in revised form:
March 13, 2003
May 27, 2003
The author was supported in part by the European Commission and EPSRC. The author would like to thank Shaun Bullett for many enlightening conversations. Also, thanks to the referee for useful comments
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