Asymptotic behavior of the maximal systole

Determine the asymptotic behavior, as the genus g tends to infinity, of the maximum systole length among closed hyperbolic surfaces of genus g.

Background

For each genus g, the systole function on the moduli space of closed hyperbolic surfaces attains a maximum, and a standard area argument gives an upper bound of approximately 2 log g. Existing constructions provide lower bounds for the limsup and liminf of the maximal systole normalized by log g, but do not determine its full asymptotic behavior.

The paper improves the known every-genus lower bound for the normalized liminf to 1 by constructing, for every sufficiently large genus g, a closed hyperbolic surface with systole at least log g minus a lower-order term. This still leaves the asymptotic behavior of the maximum itself unresolved.