Coverings of Groups, Regular Dessins, and Surfaces

TL;DR

This paper introduces a coset geometry approach to regular dessins and surfaces, characterizes face-quasiprimitive dessins as coverings of unicellular dessins, and explores coverings of simple groups, leading to new regular dessins.

Contribution

It provides a new geometric framework for understanding regular dessins and classifies face-quasiprimitive dessins as specific coverings, also initiating the study of Schur coverings of simple groups.

Findings

Three types of face-quasiprimitive regular dessins as coverings of unicellular dessins.

New constructions of regular dessins from these classifications.

Investigation of coverings between SL(2,p) and PSL(2,p) leading to Fibonacci dessins.

Abstract

A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins as coverings of unicellular regular dessins. It shows that there are exactly three O'Nan-Scott-Praeger types of face-quasiprimitive regular dessins which are smooth coverings of unicellular regular dessins, leading to new constructions of interesting families of regular dessins. Finally, a problem of determining smooth Schur covering of simple groups is initiated by studying coverings between $\SL(2,p)$ and $\PSL(2,p)$, giving rise to interesting regular dessins like Fibonacci coverings.