Klein's ten planar dessins of degree 11, and beyond

TL;DR

This paper revisits Klein's degree 11 dessins d'enfants using modern methods, extends results to prime degrees of specific types, and explores the infinite occurrence of certain prime degrees linked to group theory conjectures.

Contribution

It reinterprets Klein's work with dessins d'enfants, extends analysis to new prime degrees, and investigates the infinite occurrence of primes related to specific group structures.

Findings

Determination of passports and monodromy groups for many dessins.

Construction of topologically and geometrically correct dessins in small cases.

Support for a conjecture on infinitely many primes of specific forms and associated groups.

Abstract

We reinterpret ideas in Klein's paper on transformations of degree $11$ from the modern point of view of dessins d'enfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman-Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for some prime power $q$, in which case infinitely many groups ${\rm PSL}_n(q)$ arise as permutation groups and monodromy groups of degree $p$ (an open problem in group theory).