A classification of regular maps with Euler characteristic $-p^4$ for a prime $p\geq 5$
Xiaogang Li, Yao Tian
TL;DR
The paper advances the classification of regular maps by their Euler characteristic, focusing on characteristic $-p^4$ with primes $p\ge5$. It develops a clean group-theoretic framework to bound Sylow $p$-subgroups and uses an inductive reduction via normal $p$-subgroups to restrict possible automorphism groups. The core result is an explicit classification for $-p^4$ in terms of reduced presentations when $p\in\{5,7,13\}$, and a complete nonexistence statement for all other primes: a closed surface with Euler characteristic $-p^4$ supports no regular maps unless $p\in\{5,7,13\}$. This yields concrete spectra of regular maps for these primes and confirms the broader nonexistence for other primes, contributing a solid step toward a full inductive characterization of regular maps with Euler characteristic $-p^i$. The approach hinges on the interplay between Hurwitz-type bounds, almost Sylow-cyclic theory, and explicit cohomological constructions of automorphism groups.
Abstract
A map is a cellular decomposition of a closed surface. In the framework of classifying all regular maps by their supporting surface, it is an open problem to find all closed surfaces that support no regular maps. Classification of regular maps on surfaces with Euler characteristic $-p, -p^2, -p^3, -2p,$ and $-3p$ has already been done by several authors in a series of papers, which also show that surfaces with these Euler characteristic support no regular maps if the corresponding prime $p$ satisfies certain conditions. In this paper, assuming that $p\geq 5$ is a prime and $i\geq 4$, we show that the order of a Sylow $p$-subgroup of a regular map with Euler characteristic $-p^i$ is bounded by $p^{i-1}$ unless $p\in \{5, 7, 13\}$, and we show the existence of a normal $p$-subgroup for these regular maps whenever a Sylow $p$-subgroup has order at least $\sqrt{p^i}$, laying a solid foundation for using an inductive method to completely characterize regular maps of Euler characteristic $-p^i$. Based on this, we classify all regular maps with Euler characteristic $-p^4$ for a prime $p\geq 5$ in terms of reduced presentations of their automorphism groups. Consequently, a closed surface with Euler characteristic $-p^4$ supports no regular maps if and only if $p\notin \{2,3,5,7,13\}$.