Orientable regular maps with Euler characteristic divisible by few primes

TL;DR

This paper investigates the structure of orientable regular maps with Euler characteristic divisible by few primes, revealing connections to Lie type groups, classifying prime power cases, and providing explicit classifications for certain almost simple groups.

Contribution

It establishes new links between prime graphs and Euler characteristic divisibility, classifies groups with prime power Euler characteristic, and constructs infinite families of such groups.

Findings

Non-abelian simple composition factors are Lie type groups with rank ≤ number of prime divisors of Euler characteristic.

Complete classification of almost simple groups with prime power Euler characteristic.

Identifies all almost simple groups with Euler characteristic as a product of two prime powers, excluding some classical groups.

Abstract

Let $G$ be a $(2,m,n)$-group and let $x$ be the number of distinct primes dividing $\chi$, the Euler characteristic of $G$. We prove, first, that, apart from a finite number of known exceptions, a non-abelian simple composition factor $T$ of $G$ is a finite group of Lie type with rank $n\leq x$. This result is proved using new results connecting the prime graph of $T$ to the integer $x$. We then study the particular cases $x=1$ and $x=2$. We give a general structure statement for $(2,m,n)$-groups which have Euler characteristic a prime power, and we construct an infinite family of these objects. We also give a complete classification of those $(2,m,n)$-groups which are almost simple and for which the Euler characteristic is a prime power (there are four such). Finally we study those $(2,m,n)$-groups which are almost simple and for which the Euler characteristic is a product of two prime powers. All such groups which are not isomorphic to $PSL_2(q)$ or $PGL_2(q)$ are completely classified.