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Regularity properties of Macbeath-Hurwitz and related maps and surfaces

Gareth A. Jones

TL;DR

The paper develops a detailed number-theoretic and group-theoretic analysis of Macbeath–Hurwitz maps of type $\{3,7\}$, establishing a density-theoretic picture for inner versus outer regularity across primes via Hall’s criterion and density theorems. It generalizes the criterion to maps of type $\{3,n\}$, derives explicit parity results, and connects prime splitting behavior to the regularity of maps through precise Galois-group analyses of the associated trace-polynomials. The work combines triangle-group theory, Hurwitz surfaces, and powerful arithmetic statistics (Frobenius and Chebotarev) to predict how often inner vs outer regular maps occur, with substantial corroboration from Conder–Potočnik map databases and computational data. These results illuminate the arithmetic geometry underpinning highly symmetric Riemann and Klein surfaces and extend the framework to a broad family of triangular maps, offering both theoretical insight and practical benchmarks for further study.

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.

Regularity properties of Macbeath-Hurwitz and related maps and surfaces

TL;DR

The paper develops a detailed number-theoretic and group-theoretic analysis of Macbeath–Hurwitz maps of type , establishing a density-theoretic picture for inner versus outer regularity across primes via Hall’s criterion and density theorems. It generalizes the criterion to maps of type , derives explicit parity results, and connects prime splitting behavior to the regularity of maps through precise Galois-group analyses of the associated trace-polynomials. The work combines triangle-group theory, Hurwitz surfaces, and powerful arithmetic statistics (Frobenius and Chebotarev) to predict how often inner vs outer regular maps occur, with substantial corroboration from Conder–Potočnik map databases and computational data. These results illuminate the arithmetic geometry underpinning highly symmetric Riemann and Klein surfaces and extend the framework to a broad family of triangular maps, offering both theoretical insight and practical benchmarks for further study.

Abstract

The Macbeath-Hurwitz maps of type , obtained from the Hurwitz groups found by Macbeath, are fully regular by a result of Singerman, with automorphism group or . Hall's criterion determines which of these two properties, called inner and outer regularity, has. Inner (but not outer) regular maps yield non-orientable regular maps of the same type with automorphism group . If for a prime or mod~ the unique map is inner regular if and only if mod~. If for a prime mod~ there are three maps ; we use the density theorems of Frobenius and Chebotarev to show that in this case the sets of such primes for which or of them are inner regular have relative densities , , and respectively. Hall's criterion and its consequences are extended to the analogous Macbeath maps of type obtained from for all ; theoretical predictions on their number and properties are supported by evidence from the map databases of Conder and Poto\v cnik.
Paper Structure (17 sections, 9 theorems, 71 equations, 3 figures, 1 table)

This paper contains 17 sections, 9 theorems, 71 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Triangle groups and regular maps
  3. Hurwitz surfaces, groups and maps
  4. Inner and outer regularity
  5. Density theorems
  6. Application to $f_1$
  7. Application to $f_2$
  8. An alternative approach
  9. Macbeath maps of type $\{3,n\}$
  10. Extension of Hall's Criterion
  11. Parity
  12. Examples with $n$ odd
  13. Examples with $n$ even
  14. Galois groups and densities
  15. Generalisation to other types
  16. ...and 2 more sections

Key Result

Theorem 1.1

The group $G={\rm PSL}_2(q)$ is a Hurwitz group if and only if

Figures (3)

  • Figure 1: Some subfields of $K_2$ and the corresponding subgroups of $G_2$.
  • Figure 2: Some primitive $n$-th roots of $1$, for $n=7$, $9$ or $11$.
  • Figure 3: Some primitive $2n$-th roots of $1$, for even $n>6$.

Theorems & Definitions (9)

  • Theorem 1.1: Macbeath
  • Theorem 1.2: Hall's Criterion
  • Theorem 1.3: Density Theorem for Macbeath--Hurwitz maps
  • Theorem 1.4: Hall's Parity Theorem
  • Corollary 1.5
  • Theorem 1.6: Extended Regularity Criterion
  • Theorem 1.7: Extended Parity Theorem
  • Proposition 4.1: Theorem 2.10 of Hall77
  • Lemma 14.1