We determine optimal inequalities for the systole of all hyperbolic compact surfaces of caracteristic -1. First, we study the geometry and topology of these surfaces. Then, we describe the action of modular groups on Teichm\"{ u}ller spaces. Finaly, we give cell decompositions of fundamental domains such as the set of systoles is constant over a cell. Other optimal inequalities for other metric invariants are also given.
Categories: math.DG math.GT