Logarithmic systolic growth for hyperbolic surfaces in every genus

Abstract

More than thirty years ago, Brooks [J. Reine Angew. Math. 390 (1988), pp. 117–129] and Buser–Sarnak [Invent. Math. 117 (1994), pp. 27–56] constructed sequences of closed hyperbolic surfaces with logarithmic systolic growth in the genus. Recently, Liu and Petri [Random surfaces with large systoles, https://arxiv.org/abs/2312.11428, 2023] 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.