The number of groups of a given genus

Abstract

It is shown that the number of groups with a given genus greater than one is finite. The proof depends heavily on V. K. Proulx’s classification of groups of genus one. The key observation is that as the number of vertices of a graph imbedded on a given surface increases, the average face size of the imbedding approaches the average face size of a toroidal imbedding. The result appears to be related to Hurwitz’s theorem bounding the order of a group of conformal automorphisms on a Riemann surface of genus g.