Hyperbolic punctured spheres without arithmetic systole maximizers

TL;DR

This paper establishes bounds on the shortest essential curves in arithmetic punctured spheres with 4 to 12 cusps, revealing that for certain cusp counts, the maximal systole length is not achieved by arithmetic surfaces.

Contribution

It introduces a new method linking planar triangulations to systole bounds, extending known results for specific cusp counts and identifying cases where arithmetic surfaces are not systole maximizers.

Findings

Bounds for systole lengths for n=4 to 12 cusps

Arithmetic surfaces do not maximize systole length for n=7,10,11

Method uses correspondence between surfaces and planar triangulations

Abstract

We find bounds for the length of the systole -- the shortest essential, non-peripheral closed curve -- for arithmetic punctured spheres with $n$ cusps, for $n=4$ through $n=12$, some of which were previously known due to Schmutz. This is shown using a correspondence between such surfaces and planar triangulations. We show that for $n=7,10,11$, arithmetic surfaces do not achieve the maximal systole length.