2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75951We 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 Teichmüller 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.37 pages, 19 figuresDifferential GeometryGeometric Topologytext