  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.
