Papers on Fuchsian Functions

by John Stillwell I. General Reaarb, Poincares papers on Fuchsian and Kleinian I1OUps are of Ileat interest from at least two points of view: history, of course, but also as an inspiration for further mathematical prolless. The papers are historic as the climax of the ceometric theory of functions initiated by Riemann, and ideal representatives of the unity between analysis, ceometry, topololY and alcebra which prevailed during the 1880s. The rapid mathematical prOlless of the 20th century has been made at the expense of unity and historical perspective, and if mathematics is not to disintellate altogether, an effort must sometime be made to find its, main threads and weave them tocether 81ain. Poincares work is an excellent example of this process, and may yet prove to be at the core of a . new synthesis. Certainly, we are now able to gather up, some of the loose ends in Poincare, and a broader synthesis seems to be actually taking place in the work of Thurston. The papers I have selected include the three Ileat memoirs in the first volumes of Acta Math. -tice, on- Fuchsian groups, Fuchsian, functions, and Kleinian groups (Poincare [1882 a, b,1883]). These are the papers which made his reputation and they include many results and proofs which are now standard. They are preceded by an, unedited memoir written by Poincare in May 1880 at the height of his, creative ferment.