Non-orientable regular maps with negative prime-power Euler characteristic

TL;DR

This paper classifies all non-orientable regular maps on surfaces with Euler characteristic equal to a negative odd prime power, detailing their automorphism groups and dividing them into distinct families based on group structure.

Contribution

It provides a comprehensive classification of such maps, extending previous work for cases where the exponent is 1 or 2, and describes their automorphism groups in detail.

Findings

18 families of regular maps identified

Automorphism groups characterized as 2-groups or PSL(2,q)/PGL(2,q)

Conditions on parameters for each family established

Abstract

In this paper we provide a classification of all regular maps on surfaces of Euler characteristic $-r^d$ for some odd prime $r$ and integer $d\ge 1$. Such maps are necessarily non-orientable, and the cases where $d = 1$ or $2$ have been dealt with previously. This classification splits naturally into three parts, based on the nature of the automorphism group $G$ of the map, and particularly the structure of its quotient $G/O(G)$ where $O(G)$ is the largest normal subgroup of $G$ of odd order. In fact $G/O(G)$ is isomorphic to either a $2$-group (in which case $G$ is soluble), or $\textrm{PSL}(2,q)$ or $\textrm{PGL}(2,q)$ where $q$ is an odd prime power. The result is a collection of $18$ non-empty families of regular maps, with conditions on the associated parameters.