A classification of regular maps with Euler characteristic $-pq$

TL;DR

This paper classifies regular maps with Euler characteristic -pq for distinct primes p and q greater than or equal to 5, completing the classification for several prime-related Euler characteristics and revealing the existence of specific regular maps for twin primes.

Contribution

It provides a complete classification of regular maps with Euler characteristic -pq for primes p and q, including the existence of three such maps for twin primes greater than 5.

Findings

Complete classification of regular maps with Euler characteristic -pq for primes p,q ≥ 5.

Existence of three regular maps with solvable automorphism groups for twin primes p,q > 5.

Extension of previous classifications for Euler characteristics -2p, -3p, and -p^2.

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.