TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28«, Number 1, March 1985 NILPOTENT AUTOMORPHISMGROUPS OF RIEMANNSURFACES BY REZA ZOMORRODIAN Abstract. The action of nilpotent groups as automorphisms of compact Riemann surfaces is investigated. It is proved that the order of a nilpotent group of automor- phisms of a surface of genus g > 2 cannot exceed 16(g - 1). Exact conditions of equality are obtained. This bound corresponds to a specific Fuchsian group given by the signature (0; 2,4,8). 0.0 Introduction. The study of automorphisms of Riemann surfaces has acquired a great importance from its relation with the problems of moduli and Teichmuller space. After Schwarz, who first showed that the group of automorphisms of a compact Riemann surface of genus g > 2 is finite in the late nineteenth century, fundamental results were obtained by Hurwitz [8], who obtained the best possible bound 84(g — 1) for the order of such group. About the same time Wiman [16] made a thorough study of the cases 2 < g < 6, as well as improved this bound for a cyclic group, by showing that an exact upper bound for the order of an automor- phism is 2(2g + 1). All this was done using classical algebraic geometry, without use of Fuchsian groups. There was not much movement in the subject between the early 1900s and 1961, when Macbeath [10], following up a remark of Siegel, proved that there are infinitely many values of g for which the Hurwitz bound is attained, as well as infinitely many g for which it is not attained. Macbeath used the theory of Fuchsian groups. By then it was known that every finite group can be represented as a group of automorphisms of a compact Riemann surface of some genus g > 2 (see Hurwitz [8], Burnside [1] and Greenberg [2]). The aim of the present paper is to make a fairly detailed study of nilpotent automorphism groups of a Riemann surface of genus g > 2. The groups involved are finite, by Schwarz' theorem, and since a finite nilpotent group is the product of its Sylow subgroups, the p-localization homomorphisms (which are analogous, in a way to the method of taking residues modulo p in number theory) provide a natural tool for the study of nilpotent automorphism groups. The problem which I set out to solve is to find and prove the "nilpotent" analogue of Hurwitz' theorem. Not only does this paper present a complete solution to this Received by the editors May 24, 1983 and, in revised form, May 31, 1984. 1980 Mathematics Subject Classification. Primary 20H10, 20D15, 20D45. Key words and phrases. Fuchsian groups, nilpotent automorphism groups, compact Riemann surfaces, action of groups, generators, relations, signatures, bounds, maximal order of groups. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 241 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use