Primitive Monodromy Groups of Genus at most Two

Let X be a compact, connected Riemann surface of genus g, and let ρ : X → P (C) be a covering map of degree N . Then the monodromy group Mon(X, ρ) acts transitively on the fibre of a generic point. Such a group has genus g. We are concerned with the following question. Given an abstract finite group G and a non-negative integer g, does G arise as a monodromy group of genus g? The focus in the present paper is with primitive groups of genus at most two, that is, groups which have primitive permutation representations as monodromy groups of genus two or less. For each non-negative integer g, there a finite set Eg of simple groups such that if G is a group of genus g and S is a nonabelian composition factor of G then either S is an alternating groups or S ∈ Eg. See [FM01]. Our goal is to obtain not only the explicit list of elements of Eg, g = 0, 1, 2, but also the monodromy groups G in which these sections appear. Let G = Mon(X, ρ), as above. Then G has a distinguished generating tuple x = (x1, . . . , xr) corresponding to the branch points S of ρ. (G is a homomorphic image of the fundamental group π1(P (C) \ S), and x1, . . . , xr are the images of natural generators of this group.) For present purposes, say that G is an exceptional group of genus g if, furthermore, G has a composition factor in Eg. More precisely, we speak of the exceptional triple (G,M, x) of genus g where M is a point stabilizer and x is as above. Our goal is to determine, up to natural equivalance, the complete list of exceptional triples (G,M, x) of genus g for g at most 2. For small values of g, most of the exceptional triples occur in point actions, that is, where G is an almost simple classical group and M is the stabilizer of a point in the action of G on the 1-spaces of its natural module. For that reason, our main results here concern such actions. Our analysis uses properties of the natual module, often regarded as a vector space over the prime field. Working toward explicit descriptions of E0, E1, and E2, we show here that when g is at