Orientably regular maps with Euler characteristic divisible by few primes

Let G be a (2,m, n)-group and let x be the number of distinct primes dividing χ, the Euler characteristic of G. We prove, first, that, apart from a finite number of known exceptions, a nonabelian simple composition factor T of G is a finite group of Lie type with rank n ≤ 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 announce a result pertaining to those (2,m, n)-groups which are almost simple and for which |χ| is a product of two prime powers. All such groups which are not isomorphic to PSL2(q) or PGL2(q) are completely classified. Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Orientably regular maps with Euler characteristic divisible by few primes