Graphs of Large Girth and Surfaces of Large Systole

TL;DR

This paper introduces a novel method to construct hyperbolic surfaces with large systoles, leveraging graph theory and properties of matrices in SL(2,Z), advancing understanding of geometric bounds related to surface genus.

Contribution

It presents a new construction technique for hyperbolic surfaces with systoles growing logarithmically with genus, combining graph girth and matrix count methods.

Findings

Constructed surfaces with systoles growing logarithmically with genus.

Established bounds on matrices in SL(2,Z) with positive entries and bounded trace.

Demonstrated the effectiveness of combining graph theory and matrix analysis for surface geometry.

Abstract

The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces. In this article we provide a new construction for a sequence of surfaces with systoles that grow logarithmically in their genera. We do this by combining a construction for graphs of large girth and a count of the number of $\mathrm{SL}_2(\mathbb{Z})$ matrices with positive entries and bounded trace.