Orientable regular maps with Euler characteristic divisible by few primes

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.