Systoles and diameters of hyperbolic surfaces

TL;DR

This paper investigates the relationship between systoles and diameters of closed hyperbolic surfaces, establishing an inequality that bounds their ratio based on genus, contributing to geometric understanding of such surfaces.

Contribution

The paper introduces a new inequality linking systole and diameter of hyperbolic surfaces, providing genus-dependent bounds on their ratio.

Findings

Systole and diameter satisfy a specific inequality.

The ratio of systole to diameter has an upper bound depending on genus.

Provides geometric bounds relevant to hyperbolic surface theory.

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.