Mating Kleinian groups isomorphic to 𝐶₂∗𝐶₅ with quadratic polynomials

Given a quadratic polynomial q : C ^ → C ^ q:\hat {\mathbb {C}}\rightarrow \hat {\mathbb {C}} and a representation G : C ^ → C ^ G:\hat {\mathbb {C}} \rightarrow \hat {\mathbb {C}} of C 2 ∗ C 5 C_2\ast C_5 in P S L ( 2 , C ) PSL(2,\mathbb {C}) satisfying certain conditions, we will construct a 4 : 4 4:4 holomorphic correspondence on the sphere (given by a polynomial relation p ( z , w ) p(z,w) ) that mates the two actions: The sphere will be partitioned into two completely invariant sets Ω \Omega and Λ \Lambda . The set Λ \Lambda consists of the disjoint union of two sets, Λ + \Lambda _+ and Λ − \Lambda _- , each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial P P . This filled Julia set contains infinitely many copies of the filled Julia set of q q . Suitable restrictions of the correspondence are conformally conjugate to P P on each of Λ + \Lambda _+ and Λ − \Lambda _- . The set Λ \Lambda will not be connected, but it can be joined up using a family C \mathcal {C} of completely invariant curves. The action of the correspondence on the complement of Λ ∪ C \Lambda \cup \mathcal {C} will then be conformally conjugate to the action of G G on a simply connected subset of its regular set.