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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.10434 |
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| _version_ | 1866911105513684992 |
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| author | Jones, Gareth A. Mačaj, Martin Širáň, Jozef |
| author_facet | Jones, Gareth A. Mačaj, Martin Širáň, Jozef |
| contents | One of the consequences of residual finiteness of triangle groups is that for any given hyperbolic triple $(\ell,m,n)$ there exist infinitely many regular hypermaps of type $(\ell,m,n)$ on compact orientable surfaces. The same conclusion also follows from a classification of those finite quotients of hyperbolic triangle groups that are isomorphic to linear fractional groups over finite fields. A non-orientable analogue of this, that is, existence of regular hypermaps of a given hyperbolic type on {\em non-orientable} compact surfaces, appears to have been proved only for {\em maps}, which arise when one of the parameters $\ell,m,n$ is equal to $2$.
In this paper we establish a non-orientable version of the above statement in full generality by proving the following much stronger assertion: for every hyperbolic triple $(\ell,m,n)$ there exists an infinite set of primes $p$ of positive Dirichlet density, such that (i) there exists a regular hypermap $\mathcal{H}$ of type $(\ell,m,n)$ on a compact non-orientable surface such that the automorphism group of $\mathcal{H}$ is isomorphic to $\PSL(2,p)$, and, moreover, (ii) the carrier compact surface of {\em every} regular hypermap of type $(\ell,m,n)$ with rotation group isomorphic to $\PSL(2,p)$ is necessarily non-orientable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-orientable regular hypermaps of arbitrary hyperbolic type Jones, Gareth A. Mačaj, Martin Širáň, Jozef Group Theory Combinatorics Primary: 05C10. Secondary: 11R45, 14H45, 20B25, 30F10, 30F50 One of the consequences of residual finiteness of triangle groups is that for any given hyperbolic triple $(\ell,m,n)$ there exist infinitely many regular hypermaps of type $(\ell,m,n)$ on compact orientable surfaces. The same conclusion also follows from a classification of those finite quotients of hyperbolic triangle groups that are isomorphic to linear fractional groups over finite fields. A non-orientable analogue of this, that is, existence of regular hypermaps of a given hyperbolic type on {\em non-orientable} compact surfaces, appears to have been proved only for {\em maps}, which arise when one of the parameters $\ell,m,n$ is equal to $2$. In this paper we establish a non-orientable version of the above statement in full generality by proving the following much stronger assertion: for every hyperbolic triple $(\ell,m,n)$ there exists an infinite set of primes $p$ of positive Dirichlet density, such that (i) there exists a regular hypermap $\mathcal{H}$ of type $(\ell,m,n)$ on a compact non-orientable surface such that the automorphism group of $\mathcal{H}$ is isomorphic to $\PSL(2,p)$, and, moreover, (ii) the carrier compact surface of {\em every} regular hypermap of type $(\ell,m,n)$ with rotation group isomorphic to $\PSL(2,p)$ is necessarily non-orientable. |
| title | Non-orientable regular hypermaps of arbitrary hyperbolic type |
| topic | Group Theory Combinatorics Primary: 05C10. Secondary: 11R45, 14H45, 20B25, 30F10, 30F50 |
| url | https://arxiv.org/abs/2508.10434 |