On orientably-regular maps of Euler characteristic $-2p^2$
Tomás Foncea E., Sebastián Reyes-Carocca}
TL;DR
The paper classifies orientably-regular maps on surfaces of Euler characteristic $-2p^2$ that admit orientation-preserving automorphism groups of order $10p^2$, for primes $p\ge 11$, by analyzing actions on genus $1+p^2$ Riemann surfaces with a conformal automorphism group of order $5p^2$. It uses triangle Fuchsian groups and surface-kernel epimorphisms tied to regular Belyi pairs to derive allowed nonabelian groups of order $5p^2$, depending on $p\bmod 5$, and then provides a complete enumeration and explicit presentations for the resulting maps and their automorphism groups across four group families. The work yields concrete counts: exactly two reflexive maps when $p\equiv -1\pmod{5}$ and exactly ten maps when $p\equiv 1\pmod{5}$ (split into two reflexive and four chiral pairs), along with descriptions of the underlying Riemann surfaces, many non-hyperelliptic, and a corollary asserting the nonexistence of genus $1+p^2$ surfaces with precisely $5p^2$ conformal automorphisms. The results deepen the connection between orientably-regular maps, Belyi theory, and automorphism classifications on high-genus surfaces.
Abstract
In this article, we study orientably-regular maps of Euler characteristic $-2p^2$ and classify those that admit a group of orientation-preserving automorphisms of order $10p^2$, where $p$ is a prime number. Along the way, we classify all compact Riemann surfaces (or complex algebraic curves) of genus $1+p^2$ endowed with a group of conformal automorphisms of order $5p^2$.