Mating Kleinian Groups Isomorphic to C2 ∗ C5 with Quadratic Polynomials

Given a quadratic polynomial q : Ĉ→ Ĉ and a representation G : Ĉ→ Ĉ of C2 ∗C5 in PSL(2,C) satisfying certain conditions, we will construct a 4 : 4 holomorphic correspondence on the sphere (given by a polynomial relation p(z,w)) that mates the two actions: The sphere will be partitioned into two completely invariant sets Ω and Λ. The set Λ consists of the disjoint union of two sets, Λ+ and Λ−, each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial P . This filled Julia set contains infinitely many copies of the filled Julia set of q. Suitable restrictions of the correspondence are conformally conjugate to P on each of Λ+ and Λ−. The set Λ will not be connected, but it can be joined up using a family C of completely invariant curves. The action of the correspondence on the complement of Λ∪C will then be conformally conjugate to the action of G on a simply connected subset of its regular set. 1. Background and motivation The theories of iterated rational maps [3], [7] and Kleinian groups [2], [10], both acting on the Riemann sphere Ĉ = C∪∞ exhibit a number of striking similarities, which arise from the fact that in both cases Ĉ is partitioned into two completely invariant sets, namely the regular set Ω and the limit Λ in the case of a Kleinian group, and the Fatou set F and the Julia set J in the case of a rational map. Orbits of points under the group or under backward iteration of the rational map accumulate on the limit or Julia set respectively, whereas the action of the group or rational map on the regular or Fatou set is discontinuous and equicontinuous. One can mate two abstractly isomorphic Fuchsian groups G1 and G2 which are topologically conjugate on the upper half plane by gluing them together at their limit sets. This is realised by a third quasi-Fuchsian group G whose regular set consists of two simply connected components. On each of these components the action of G is conformally conjugate to one of the Gi. Similarly, one can mate two hyperbolic quadratic polynomials q1 and q2 (which both lie in the main cardiod of the Mandelbrot set) via a third rational map R by gluing them together at their Julia sets. The Fatou set of R will consist of two completely invariant components, Received by the editors November 23, 2001 and, in revised form, March 13, 2003. 2000 Mathematics Subject Classification. Primary 37F45, 37F30, 37F05; Secondary 37F10.