Systoles and diameters of hyperbolic surfaces

Abstract : In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus dependent) upper bound. asymptotic question was recently settled by Budzinski, Curien and Petri [6] where they * Supported by the FSE/AEI/MICINN grant RYC-2016-19334 "Local and global systolic geometry and topology" and the FEDER/AEI/MICIU grant PGC2018-095998-B-I00 "Local and global invariants in geometry".