Hurwitz groups as monodromy groups of dessins: several examples

TL;DR

This paper demonstrates how small quotient dessins can serve as practical substitutes for larger Galois covers, using group-theoretic techniques to analyze Hurwitz groups and dessins.

Contribution

It provides new examples illustrating the use of small quotient dessins as proxies for complex Galois covers, with insights into group-theoretic methods.

Findings

Small quotient dessins effectively approximate larger Galois covers.

Group-theoretic techniques like Frobenius formula are useful in analyzing dessins.

Examples are relevant to dessins of all types, not just Hurwitz groups.

Abstract

We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.