Title: 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).