Local maxima of the systole function

Abstract

We construct a sequence of closed hyperbolic surfaces that are local maxima for the systole function in their respective moduli spaces. Their systole is arbitrarily large and the number of examples grows rapidly with the genus. More precisely, for every $n\ge 3$ there is some positive number $L_n$ (growing roughly linearly in $n$) such that the number of local maxima of the systole function in genus $g$ with systole equal to $L_n$ grows super-exponentially in $g$ along an arithmetic sequence of step size $n$. Many of these surfaces have no orientation-preserving isometries other than the identity and are the first examples of local maxima with this property.