Mating Parabolic Rational Maps with Hecke Groups

The paper’s Main Theorem states that for any degree-d rational map R with a parabolic fixed point of multiplier 1 and fully invariant, simply connected immediate basin, there is a d:d holomorphic correspondence F that is a mating with the Hecke group H_{d+1}, and moreover F = J ∘ Cov_P^0 for a degree-(d+1) polynomial P and a conformal involution J. The structure of the proof is explicitly two-step: (i) construct a pinched polynomial-like mating between R and a (piecewise real-analytic, expansive) Hecke/Farey external map, and (ii) globalize by either a double-cover construction or via B-involutions and a (d+1)-fold cover, yielding the algebraic correspondence form J ∘ Cov_P^0. These points appear verbatim in the paper’s statement and plan (Definition 1.1; Main Theorem; Sections 5 and 6), and the details include the Hecke/Farey external maps with parabolic asymptotics at 1, the pinched polynomial-like category, Corollary 5.4 for the existence of RH and RF, the double-cover method constructing a d:d correspondence, the B-involution framework (Definition 6.2), and Proposition 6.1 identifying F = J ∘ Cov_P^0. The candidate solution reproduces this proof architecture and key ingredients without introducing contradictions or unsupported claims, so both are correct and essentially the same proof. Key citations: Main Theorem and the J ∘ Cov_P^0 form (); mating definition/axioms (); two-step plan and B-involutions (); external Hecke/Farey construction and asymptotics (, ); pinched polynomial-like mating (Corollary 5.4) (); double-cover construction and J-conjugacy (, ); algebraicity and F = J ∘ Cov_P^0 (); B-involutions and the (d+1)-fold cover method (, ).