Asymptotic properties of the set of systoles of arithmetic Riemann surfaces

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).