Mating parabolic rational maps with Hecke groups

We prove that any degree rational map having a parabolic fixed point of multiplier 1 with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group , with the mating realised by an algebraic correspondence. This confirms the parabolic version of a conjecture on mateability between rational maps and Hecke groups made by Bullett and Freiberger [Int. J. Mod. Phys. B 17 (2003), 3922–3931]. The proof is in two steps. The first is the construction of a pinched polynomial‐like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial‐like map to an algebraic correspondence via a suitable branched covering.