A classification of regular maps with Euler characteristic $-p^4$ for a prime $p\geq 5$

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 $p\notin \{2,3,5,7,13\}$.