Maximal order group actions on Riemann surfaces
Keywords:Riemann surface, genus, group action, NEC group, strong symmetric genus
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) ≤ 2N(g). There are well-known families of extended Hurwitz groups that provide an infinite number of integers g satisfying M(g) = 2N(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 = 3p + 1 or g = 2p + 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.