Systoles and kissing numbers of finite area hyperbolic surfaces

TL;DR

This paper investigates the maximum number and lengths of systoles on finite area hyperbolic surfaces, providing bounds that depend on topology and grow subquadratically with genus.

Contribution

It establishes new upper bounds on the kissing number for hyperbolic surfaces that depend solely on surface topology and grow subquadratically with genus.

Findings

Upper bounds on the number of systoles depending on topology

Kissing number grows subquadratically with genus

Results applicable to complete finite area hyperbolic surfaces

Abstract

We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surface and which grows subquadratically in the genus.