Regularity properties of Macbeath-Hurwitz and related maps and surfaces

TL;DR

This paper investigates the regularity properties of Macbeath-Hurwitz maps of type {3,7} derived from Hurwitz groups, classifies their inner and outer regularity based on prime conditions, and extends these results to maps of type {3,n} for all n≥7.

Contribution

It provides a complete classification of the regularity types of Macbeath-Hurwitz maps of type {3,7} and generalizes the criteria to maps of type {3,n} for all n≥7, supported by density theorems and database evidence.

Findings

Inner regular maps occur under specific prime congruences.

Density theorems determine the distribution of regularity types among primes.

Extension of regularity criteria to maps of type {3,n} for all n≥7.

Abstract

The Macbeath-Hurwitz maps $M$ of type $\{3,7\}$, obtained from the Hurwitz groups $G={\rm PSL}_2(q)$ found by Macbeath, are fully regular by a result of Singerman, with automorphism group $G\times{\rm C}_2$ or ${\rm PGL}_2(q)$. Hall's criterion determines which of these two properties, called inner and outer regularity, $M$ has. Inner (but not outer) regular maps $M$ yield non-orientable regular maps $M/{\rm C}_2$ of the same type with automorphism group $G$. If $q=p^3$ for a prime $p\equiv\pm 2$ or $\pm 3$ mod~$(7)$ the unique map $M$ is inner regular if and only if $p\equiv 1$ mod~$(4)$. If $q=p$ for a prime $p\equiv\pm 1$ mod~$(7)$ there are three maps $M$; we use the density theorems of Frobenius and Chebotarev to show that in this case the sets of such primes $p$ for which $0, 1, 2$ or $3$ of them are inner regular have relative densities $1/8$, $3/8$, $3/8$ and $1/8$ respectively. Hall's criterion and its consequences are extended to the analogous Macbeath maps of type $\{3,n\}$ obtained from ${\rm PSL}_2(q)$ for all $n\ge 7$; theoretical predictions on their number and properties are supported by evidence from the map databases of Conder and Poto\v cnik.