A domain in the complex plane is called a quadrature domain if it admits a global Schwarz reflection map. Topology of quadrature domains has important applications to physics, and is intimately related to iteration of Schwarz reflection maps.

We will consider a specific one-parameter family of Schwarz reflection maps, and show that every post-critically finite map in this family arises as the mating of a post-critically finite quadratic anti-holomorphic polynomial and the ideal triangle group. Time permitting, we will also describe the connectedness locus of this family in terms of the tricorn.

Joint work with Mikhail Lyubich, and Nikolai Makarov.