Regularity properties of Macbeath-Hurwitz and related maps and surfaces
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911255006019584 |
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| author | Jones, Gareth A. |
| author_facet | Jones, Gareth A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity properties of Macbeath-Hurwitz and related maps and surfaces Jones, Gareth A. Combinatorics 05C10 (primary), 11R45 14H45, 20B25, 30F10, 30F50 (secondary) 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. |
| title | Regularity properties of Macbeath-Hurwitz and related maps and surfaces |
| topic | Combinatorics 05C10 (primary), 11R45 14H45, 20B25, 30F10, 30F50 (secondary) |
| url | https://arxiv.org/abs/2505.02089 |