The systoles of a hyperbolic surface { \Sigma} are the shortest closed geodesics. We say that the systoles fill the surface if the set Syst({ \Sigma}) of all systoles cuts { \Sigma} into polygons. We refine an idea of Schmutz [15] to construct closed hyperbolic surfaces { \Sigma} of arbitrarily large genus with a small set Syst({ \Sigma}) that fills. In fact, for the surfaces { \Sigma} considered, the cardinality of Syst({ \Sigma}) is in o(g/ ln g), where g is the genus of { \Sigma}. The proof is based on the theory Coxeter groups, combined with some elementary number theory.
Categories: math.GT math.GR