Construction of hyperbolic Riemann surfaces with large systoles

TL;DR

This paper constructs hyperbolic Riemann surfaces with large systoles using cutting and pasting techniques, establishing inequalities relating maximal systole lengths across different genera and extending results to non-compact surfaces with cusps.

Contribution

It introduces a novel construction method for hyperbolic Riemann surfaces with large systoles and derives new inequalities linking systole lengths among surfaces of varying genus.

Findings

Constructed hyperbolic surfaces with large systoles.

Established inequalities between maximal systole lengths for different genera.

Extended systolic inequalities to non-compact surfaces with cusps.

Abstract

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (globally) maximal surface $S_{max}$ a compact Riemann surface of genus $g$ whose systole has length $\mathop{msys}(g)$. In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating $\mathop{msys}(\cdot)$ of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.