Asymptotic properties of the set of systoles of arithmetic Riemann surfaces

TL;DR

This paper explores the distribution and asymptotic behavior of systoles in arithmetic hyperbolic surfaces, linking combinatorics, group theory, and geometry to understand their distribution in moduli space.

Contribution

It provides new insights into the distribution of systoles on arithmetic surfaces, showing they cannot be overly concentrated, using diverse mathematical techniques.

Findings

Systoles of arithmetic surfaces are well distributed and not concentrated.

Connections established between graph theory and surface geometry.

Results suggest asymptotic behaviors align with moduli space volume growth.

Abstract

The purpose this article is to try to understand the mysterious coincidence between the asymptotic behavior of the volumes of the Moduli Space of closed hyperbolic surfaces of genus $g$ with respect to the Weil-Petersson metric and the asymptotic behavior of the number of arithmetic closed hyperbolic surfaces of genus $g$. If the set of arithmetic surfaces is well distributed then its image for any interesting function should be well distributed too. We investigate the distribution of the function systole. We give several results indicating that the systoles of arithmetic surfaces can not be concentrated, consequently the same holds for the set of arithmetic surfaces. The proofs are based in different techniques: combinatorics (obtaining regular graphs with any girth from results of B. Bollobas and constructions with cages and Ramanujan graphs), group theory (constructing finite index subgroups of surface groups from finite index subgroups of free groups using results of G. Baumslag) and geometric group theory (linking the geometry of graphs with the geometry of coverings of a surface).