Hurwitz Groups and Surfaces
نویسنده
چکیده
Hurwitz not only gave an upper bound for the number of automorphisms of a compact Riemann surface of genus greater than 2, but also gave a characterization of which finite groups could be groups of automorphisms achieving this bound. In practice, however, the identification of such groups and of the surfaces they act on is difficult except in special cases. We survey what is known. 1. How I Got Started on Hurwitz Groups One day in the late 1950’s, rereading Siegel’s article [1945] entitled “Some remarks on discontinuous groups”, I was struck by his proof that the smallest area of fundamental region for a Fuchsian group is π/21. Siegel notes the remarkable similarity between the arithmetic used in his proof and the arithmetic in Hurwitz’s proof that a curve of genus g ≥ 2 has no more than 84(g − 1) birational self-transformations. That, he said, is not surprising because of the theory of uniformization. That was all— no indication where to find Hurwitz’s paper, at that time unknown to me. (Siegel is one of my heroes, but, it must be confessed, he was not very good at citing references.) I did know about uniformization, and I made that connection at once. However, I had some trouble tracking down Hurwitz’s theorem. Finally, thanks to the late Professor W. L. Edge, I read Hurwitz’s paper [1893], which invoked Klein’s surface as an example to show that his bound was attained. So at last, by a very tortuous path, I unearthed this chapter of mathematics, which has fascinated me ever since. Hurwitz left open the question whether there was any other surface with the maximum number 84(g − 1) of automorphisms, as we now call them. Only one other such surface was found, by Fricke, in the sixty years to 1961. My own first contribution [Macbeath 1961] was a proof that there are infinitely many of them. My research changed direction when I became aware of Klein’s curve and Hurwitz’s theorem. I was driven to think more and more about Riemann surfaces with many automorphisms. It was natural to progress to Riemann surfaces in
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