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The purpose this article is to try to understand the mysterious coincidencebetween the asymptotic behavior of the volumes of the Moduli Space of closedhyperbolic surfaces of genus $g$ with respect to the Weil-Petersson metric andthe asymptotic behavior of the number of arithmetic closed hyperbolic surfacesof genus $g$. If the set of arithmetic surfaces is well distributed then itsimage for any interesting function should be well distributed too. Weinvestigate the distribution of the function systole. We give several resultsindicating that the systoles of arithmetic surfaces can not be concentrated,consequently the same holds for the set of arithmetic surfaces. The proofs arebased in different techniques: combinatorics (obtaining regular graphs with anygirth from results of B. Bollobas and constructions with cages and Ramanujangraphs), group theory (constructing finite index subgroups of surface groupsfrom finite index subgroups of free groups using results of G. Baumslag) andgeometric group theory (linking the geometry of graphs with the geometry ofcoverings of a surface).

Asymptotic properties of the set of systoles of arithmetic Riemann surfaces