Orientably regular maps with Euler characteristic divisible by few primes

Gill, Nick (2013). Orientably regular maps with Euler characteristic divisible by few primes. Journal of the London Mathematical Society, 88(1) pp. 118–136.

DOI: https://doi.org/10.1112/jlms/jdt010

Abstract

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 non- abelian 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.

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