1. How I Got Started on Hurwitz Groups

One day in the late 1950's, rereading Siegel's article[1945] entitled “Some remarks on discontinuousgroups”, I was struck by his proof that the smallestarea of fundamental region for a Fuchsian group isπ /21. Siegel notesthe remarkable similarity between the arithmeticused in his proof and the arithmetic in Hurwitz'sproof that a curve of genus g ≥ 2 has no more than 84(g - 1) birationalself-transformations. That, he said, is notsurprising because of the theory of uniformization.That was all- no indication where to find Hurwitz'spaper, at that time unknown to me. (Siegel is one ofmy heroes, but, it must be confessed, he was notvery good at citing references.) I did know aboutuniformization, and I made that connection at once.However, I had some trouble tracking down Hurwitz'stheorem. Finally, thanks to the late Professor W. L.Edge, I read Hurwitz's paper [1893], which invokedKlein's surface as an example to show that his boundwas attained. So at last, by a very tortuous path, Iunearthed this chapter of mathematics, which hasfascinated me ever since.