Logarithmic systolic growth for hyperbolic surfaces in every genus

TL;DR

This paper demonstrates that all hyperbolic surfaces, regardless of genus, exhibit logarithmic growth in systolic length, extending previous results through a direct approach based on classical constructions.

Contribution

It provides a new, more direct proof that hyperbolic surfaces in every genus have logarithmic systolic growth, building on classical surface constructions.

Findings

Logarithmic systolic growth holds for all genera

A new proof method based on original Brooks/Buser-Sarnak surfaces

Extends previous results to all hyperbolic surfaces

Abstract

More than thirty years ago, Brooks and Buser-Sarnak constructed sequences of closed hyperbolic surfaces with logarithmic systolic growth in the genus. Recently, Liu and Petri showed that such logarithmic systolic lower bound holds for every genus (not merely for genera in some infinite sequence) using random surfaces. In this article, we show a similar result through a more direct approach relying on the original Brooks/Buser-Sarnak surfaces.