Correspondences on hyperelliptic surfaces, combination theorems, and Hurwitz spaces

TL;DR

This paper develops a framework for constructing and analyzing correspondences on hyperelliptic surfaces by combining Fuchsian groups and Blaschke products, linking moduli spaces with Hurwitz spaces.

Contribution

It introduces a new class of correspondences on hyperelliptic surfaces, combining Fuchsian groups and Blaschke products, with an explicit moduli space description and a natural injection into Hurwitz spaces.

Findings

Constructed a general class of correspondences on hyperelliptic surfaces.

Established an algebraic characterization using involutions and meromorphic maps.

Identified the moduli space with a product of Teichmüller and Blaschke spaces.

Abstract

We construct a general class of correspondences on hyperelliptic Riemann surfaces of arbitrary genus that combine finitely many Fuchsian genus zero orbifold groups and Blaschke products. As an intermediate step, we first construct analytic combinations of these objects as partially defined maps on the Riemann sphere. We then give an algebraic characterization of these analytic combinations in terms of hyperelliptic involutions and meromorphic maps on compact Riemann surfaces. These involutions and meromorphic maps, in turn, give rise to the desired correspondences. The moduli space of such correspondences can be identified with a product of Teichm\"uller spaces and Blaschke spaces. The explicit description of the correspondences then allows us to construct a dynamically natural injection of this product space into appropriate Hurwitz spaces.