A classification of regular maps with Euler characteristic $-pq$
Xiaogang Li, Yao Tian
TL;DR
This work resolves the classification of regular maps on surfaces with Euler characteristic $\chi = -pq$ for distinct primes $p<q$ with $q>p\ge 5$, completing the program for several related Euler characteristics. The authors develop an algebraic-map framework and construct explicit map families ${\cal M}_1(j,k)$, ${\cal M}_2(x,n,p)$, ${\cal M}_3(u)$ whose automorphism groups are constrained to be almost Sylow-cyclic or PSL$(2,f)$/PGL$(2,f)$-type, and then perform a case-splitting by the divisibility of $|G|$ by $pq$. The main result enumerates all possible ${\cal M}$ (up to duality and isomorphism) into these families and, in the solvable twin-prime corollary, shows the existence of three regular maps with Euler characteristic $-pq$ for twin primes $p,q>5$. The methods connect topological, group-theoretical, and geometric aspects of regular maps, providing a comprehensive topological classification with concrete group-theoretic realizations and implications for infinite families under a twin-prime conjecture.
Abstract
In this paper, we give a classification of regular maps with Euler characteristic $-pq$ for distinct primes $q>p\geq 5$. This together with previous classification of regular maps with Euler characteristic $-2p,-3p$ and $-p^2$ completes the classification of regular maps with Euler characteristic $-pq$ for two primes $p$ and $q$. An interesting consequence is that, for every pair of twin primes $p$ and $q$ greater than $5$, there exist three regular maps with solvable automorphism groups and Euler characteristic $-pq$, up to duality and isomorphism.