Hypermaps and multiply quasiplatonic Riemann surfaces

TL;DR

This paper uses finite group theory to construct Riemann surfaces with multiple regular dessins, often sharing automorphism groups, generalizing previous examples and expanding understanding of hypermaps and quasiplatonic surfaces.

Contribution

It introduces a method to construct Riemann surfaces with multiple regular dessins having automorphism groups of the same order, extending prior work by Girondo and Wolfart.

Findings

Constructed new examples of Riemann surfaces with multiple regular dessins

Demonstrated cases with isomorphic automorphism groups

Generalized previous specific examples to broader classes

Abstract

Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases with isomorphic automorphism groups.