Introduction. Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmuller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that

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... is throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(#-1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL{2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3-1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families. Riemann surfaces and NEC groups. By a Riemann surface in this paper we shall mean a surface S together with an open covering by a family of sets fy = {t/J with the properties 1. For each U^ty there exists a homeomorphism f : i/j -» C, where C is the complex plane. 2. If U h UjB^l and i/jOf/j # 0 then ^>i» S is called an automorphism if , -o / oJ 1 is either a conformal or anticonformal mapping in its domain of definition. If 5 is an orientable surface and / i s orientation preserving (reversing) t h e n / i s called a + automorphism ( -automorphism). (In the papers of Macbeath, Lehner and Newman etc., their automorphisms are + automorphisms.) +Automorphisms of orientable Riemann surfaces have been studied by means of Fuchsian groups. We shall study automorphisms of non-orientable Riemann surfaces by means of the non-Euclidean crystallographic (NEC) groups introduced by Wilkie [7]. Let D denote the upper-half complex plane and ^ the group of conformal and anticonformal homeomorphisms of D. The elements of ^ are the transformations of the form (i) z -» a, b, c, d real, ad-be = 1, cz + d (ii) z -> --a, b, c, d real, ad-be = -1. cz + d The elements of type (i), the conformal homeomorphisms, form a subgroup of index 2 in which we denote by ^+ . https://www.cambridge.org/core/terms. https://doi.