Mating quadratic maps with Kleinian groups via quasiconformal surgery

Let $q:\hat{\mathbb C} \to \hat{\mathbb C}$ be any quadratic polynomial and $r:C_2*C_3 \to PSL(2,{\mathbb C})$ be any faithful discrete representation of the free product of finite cyclic groups $C_2$ and $C_3$ (of orders $2$ and $3$) having 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 $p(z,w)=0$.