Mating Kleinian groups isomorphic to $C_2\ast C_5$ with quadratic polynomials

Abstract:

Given a quadratic polynomial $q:\hat {\mathbb {C}}\rightarrow \hat {\mathbb {C}}$ and a representation $G:\hat {\mathbb {C}} \rightarrow \hat {\mathbb {C}}$ of $C_2\ast C_5$ in $PSL(2,\mathbb {C})$ 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 $\Omega$ and $\Lambda$. The set $\Lambda$ consists of the disjoint union of two sets, $\Lambda _+$ and $\Lambda _-$, 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 $\Lambda _+$ and $\Lambda _-$. The set $\Lambda$ will not be connected, but it can be joined up using a family $\mathcal {C}$ of completely invariant curves. The action of the correspondence on the complement of $\Lambda \cup \mathcal {C}$ will then be conformally conjugate to the action of $G$ on a simply connected subset of its regular set.