Systole growth for finite area hyperbolic surfaces

TL;DR

This paper investigates the maximum systole length on finite area hyperbolic surfaces, showing it increases with the number of cusps and exceeds a logarithmic growth bound related to genus and cusps.

Contribution

It establishes new monotonicity properties of the systole function and provides bounds relating systole length to surface parameters.

Findings

Maximum systole increases with the number of cusps for small n

Systole length exceeds a logarithmic function of g/n

Provides bounds on systole growth for hyperbolic surfaces

Abstract

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that grows logarithmically in function of the ratio $g/n$.