The homology systole of hyperbolic Riemann surfaces

TL;DR

This paper investigates the relationship between maximum length systoles and homological systoles on hyperbolic Riemann surfaces, establishing equivalences for certain surface types and identifying limitations for surfaces with many cusps.

Contribution

It demonstrates that maximizing the length systole is equivalent to maximizing the length homological systole for closed and once-punctured hyperbolic surfaces, but not for surfaces with many cusps.

Findings

Maximum length systole and homological systole coincide for closed surfaces.

The equivalence extends to once-punctured surfaces.

The equivalence fails for surfaces with many cusps.

Abstract

The main goal of this note is to show that the study of closed hyperbolic surfaces with maximum length systole is in fact the study of surfaces with maximum length homological systole. The same result is shown to be true for once-punctured surfaces, and is shown to fail for surfaces with a large number of cusps.