The separating systole and the genus ratio of high genus triangulations

TL;DR

This paper investigates the properties of high genus triangulations, demonstrating that their separating systole grows logarithmically with size and establishing the convergence of the genus ratio, extending previous size ratio results.

Contribution

It introduces new bounds on the separating systole and proves the convergence of the genus ratio in high genus triangulations, complementing earlier size ratio findings.

Findings

Separating systole is logarithmic in triangulation size

Genus ratio converges for high genus triangulations

Extends previous size ratio convergence results

Abstract

We show that the separating systole of high genus triangulations is of logarithmic order (in the size of the triangulation). Our methods also allow us to show an enumerative result, i.e. the convergence of the "genus ratio" for high genus triangulations. This complements the convergence of the "size ratio" that was proven in previous work with Budzinski.