Non-orientable regular maps with negative prime-power Euler characteristic
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arXiv
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| Format: | Preprint |
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2025
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| author | Conder, Marston Gill, Nick Širáň, Jozef |
| author_facet | Conder, Marston Gill, Nick Širáň, Jozef |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03667 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-orientable regular maps with negative prime-power Euler characteristic Conder, Marston Gill, Nick Širáň, Jozef Group Theory Combinatorics 05E18, 20B25, 20F05 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. |
| title | Non-orientable regular maps with negative prime-power Euler characteristic |
| topic | Group Theory Combinatorics 05E18, 20B25, 20F05 |
| url | https://arxiv.org/abs/2507.03667 |