SCHWARZ REFLECTIONS AND THE TRICORN

We continue our exploration of the family S of Schwarz reflection maps with respect to the cardioid and a circle which was initiated in our earlier work. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials. We also show that every geometrically finite map in S arises as a conformal mating of a unique geometrically finite quadratic anti-holomorphic polynomial and a reflection map arising from the ideal triangle group. We then follow up with a combinatorial mating description for the periodically repelling maps in S. Finally, we show that the locally connected topological model of the connectedness locus of S is naturally homeomorphic to such a model of the basilica limb of the Tricorn.