Small Systle Sets and Coxeter Groups

Abstract

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