Maximal order group actions on Riemann surfaces

Abstract

A natural problem is to determine, for each value of the integer g ≥ 2 , the largest order of a group that acts on a Riemann surface of genus g . Let N ( g ) (respectively M ( g ) ) be the largest order of a group of automorphisms of a Riemann surface of genus g ≥ 2 preserving the orientation (respectively possibly reversing the orientation) of the surface. The basic inequalities comparing N ( g ) and M ( g ) are N ( g ) ≤ M ( g ) ≤ 2 N ( g ) . There are well-known families of extended Hurwitz groups that provide an infinite number of integers g satisfying M ( g ) = 2 N ( g ) . It is also easy to see that there are solvable groups which provide an infinite number of such examples. We prove that, perhaps surprisingly, there are an infinite number of integers g such that N ( g ) = M ( g ) . Specifically, if p is a prime satisfying p ≡ 1 (mod 6) and g = 3 p + 1 or g = 2 p + 1 , there is a group of order 24( g − 1) that acts on a surface of genus g preserving the orientation of the surface. For all such values of g larger than a fixed constant, there are no groups with order larger than 24( g − 1) that act on a surface of genus g .