No abstract

This paper deals with the algebraic classification of non-euclidean plane crystallographic groups (NEC groups, for short) with compact quotient space. The groups considered are the discrete groups of motions of the Lobatschewsky or hyperbolic plane, including those which contain orientation-reversing reflections and glide-reflections. The corresponding problem for Fuchsian groups, which contain only orientable transformations, is essentially solved in the work of Fricke and Klein (6).

By a theorem of Hurwitz [3], an algebraic curve of genus g ≧ 2 cannot have more than 84(g − l) birational self-transformations, or, as we shall call them, automorphisms. The bound is attained for Klein's quarticof 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(g − 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 theory of Riemann surfaces with biholomorphic transformations into themselves, or, what is essentially the same thing, curves with birational self-transformations, was begun very long ago. Klein's quartic curve (6) of genus 3 with its' group of 168 automorphisms (as we shall call birational self-transformations) was the object of much study, and Klein's discovery is commemorated by the name 'Klein's group' for the simple group of order 168, though the group, as LF(2,7), must certainly have been known to Galois.When g = 3, 84(gr -1) = 168, and Hurwitz, in his fundamental paper (5), showed that a Riemann surface of genus g ^ 2 cannot have more than 84(gr -1) automorphisms, and he gave a simple group-theoretic criterion for a finite group to occur as a group of 84(gr-1) automorphisms of a curve of genus g. However, this simple and direct lead was not followed Up until 1961, when I inferred the existence of infinitely many curves for which the Hurwitz bound is attained (10). Using the methods of classical algebraic geometry, Wiman (12) had in the meantime obtained fairly exhaustive results about genus g = 2, 3, 4, 5, and 6, establishing in particular that only for g = 3 is the maximum Si(g-1) attained. Had he gone one step further, to genus 7, he, would have found a curve whose group of 504 automorphisms is isomorphic to LF(2,2 3 ). This curve is the subject of the present paper.Indeed the existence of such a curve is immediately deducible from Hurwitz's criterion and the presentation of the group LF(2,2 3 ) given in Coxeter and Moser's book ((3) 97). However, Hurwitz's method only tells us how to obtain the Riemann surface of the curve as an abstract Riemann surface, and does not give any algorithm for obtaining equations for the curve. In general, no algorithm is known for deriving equations of curves with known groups of automorphisms, but if the automorphism group is soluble, and the field of invariant functions is the field of a rational curve, then, by the theory of Galois, the field of the curve is an extension by radicals, and if enough is known about the branch-points this will enable us to obtain equations defining the curve. Now the group LF(2,2 3 ) which we are considering has a soluble subgroup of Proe. London Math. Soc. (3) IS (1965) 527-42 528 A. M. MACBEATH

1. THE problem treated here is an example of a wide class of problems involving a space E which has (i) a binary law of composition, e.g. addition, group multiplication; (ii) a real-valued monotonic set-function /x.Let the law of composition be denoted by o. Then, HA,B are two subsets of E, we define AoB = {xoy\zeA,ye B}. Our problem is: given fx{A), fi (B), what can be said about the value of fi{A oB)?Well-known results of this type are the (a-f/3)-theorem, first proved by H. B. Mann (10), and the analogous Cauchy-t)avenport theorem on finite cyclic groups (4, 5). In the first of these ft is the density of the set, in the second it is simply the number of elements.In the present paper we deal with Euclidean n-space, /* being the Lebesgue measure, and the composition law being vector addition. Our inequality was first proved by Brunn (3) for convex sets, and conditions for equality to hold were added by Minkowski (11). In 1&35 Lusternik (9) generalized the result to arbitrary measurable sets and stated conditions for equality which, as we shall see, are correct, though Lusternik's proof of them is defective. A certain amount of recent literature has appeared applying the result to the isoperimetric problem, and, in particular, mention should be made of (13), E. Schmidt's work generalizing certain restricted forms of the result to non-Euclidean geometries. See also the paper by Hadwiger (6).Lusternik's proof is open to the general criticism that, although he states the result for arbitrary measurable sets, he is inclined to assume that these sets have special properties, e.g. that their intersections with linear subspaces of E n are measurable in the lower dimension. This is false, in general, but the difficulty can easily be surmounted by assuming the sets to be J^-sets (as we do), and then deducing the more general result from the principle that every measurable set contains an F a -Bot of the same measure. Thus Lusternik's proof of the inequality is essentially valid.There is a more serious fault in his proof of the conditions for equality. At (9), p. 57,11. 30 sqq., he is considering the situation of a hyperplane M o . Ifp e M o , he denotes by L^ the line through p perpendicular to M Q . A and B are assumed to be two sets of finite positive measure, and it is supposed