On a Theorem of Hurwitz

of genus 3 [4]. In studying the problem whether there are any other curves for which the bound is attained, I was led to consider the universal covering space of the Riemann surface, which, as Siegel observed, relates Hurwitz's theorem to Siegel's own result [7] on the measure of the fundamental region of Fuchsian groups. Any curve with 84(gr — 1) automorphisms must be uniformized by a normal subgroup of the triangle group (2, 3, 7), and, by a closer analysis of possible finite factor groups of (2, 3, 7), purely algebraic methods yield an infinite family of curves with the maximum number of automorphisms. This will be shown in a later paper. The aim of the present paper, however, is to show how one can prove the existence of such an infinite family without any knowledge of the algebraic structure of the triangle group. All one needs is a single curve with the maximum number of automorphisms—and for this we may use Klein's quartic. By lifting the group of automorphisms to the universal covering space we obtain an extension of the fundamental group of the curve. If we then collapse the fundamental group to the first homology group modulo m—which involves factoring out a normal subgroup of finite index—we arrive at a finite extension of the original group and a finite-sheeted covering space of the original Riemann surface. Thus, if g is the genus of the original curve, we obtain, for every integer m, a curve with the maximum number of automorphisms and genus (g — \)m+1. It is possible to construct an extension group of this kind for more general spaces, and the main part of the paper deals with this. The results belong, on the whole, to a familiar part of classical topology, but the present application requires a restatement of them which may be useful in other contexts. In what follows, we shall not necessarily be interested in the whole group of homeomorphisms of a space, but rather in those which preserve some local structure, e.g. a differentiable or analytic structure. The term " automorphism " is used as an omnibus word to cover " homeomorphism", " differentiate homeomorphism of class C r " , or "one-one analytic mapping " according to context. We do not completely specify the kinds of local structure to which the results apply—perhaps there might be applications to local affine structures or local group structures, for instance. However, the immediate application we have in mind is that of analytic mappings of a Riemann surface.