Combining cusped triangle groups with Blaschke products: commensurable matings

In this note, we construct algebraic correspondences as matings of Fuchsian $(p,q,\infty)$-triangle groups with Blaschke products.

Combined with the results of [MM25], this proves mateability of all cusped triangle groups with suitable Blaschke products.

The proof of the main result involves associating two piecewise analytic circle maps to the $(p,q,\infty)-$triangle group, mating these maps with appropriate Blaschke products to produce two commensurable conformal matings, and finally constructing the desired algebraic correspondence as a common lift of the two conformal matings.