Local maxima of the systole function

TL;DR

This paper constructs infinite families of hyperbolic surfaces that are local maxima of the systole function, revealing complex topological structures and new examples with trivial automorphism groups.

Contribution

It introduces new infinite families of hyperbolic surfaces that are local maxima of the systole function, including the first examples with trivial automorphism groups.

Findings

Existence of local maxima at specific systole lengths

Super-exponential growth of local maxima in certain genera

Level sets of systole function can have many connected components

Abstract

We construct infinite families of closed hyperbolic surfaces that are local maxima for the systole function on their respective moduli spaces. The systole takes values along a linearly divergent sequence $(L_n)_{n\geq 1}$ at these local maxima. The only surface corresponding to $L_1\approx 3.057$ is the Bolza surface in genus $2$. For every genus $g\geq 13$, we obtain either one or two local maxima in $\mathcal{M}_g$ whose systoles have length $L_2\approx 5.909$. For each $n\geq 3$, there is an arithmetic sequence of genera $(g_k)_{k\geq 1}$ such that the number of local maxima of the systole function in $\mathcal{M}_{g_k}$ at height $L_n$ grows super-exponentially in $g_k$. In particular, level sets of the systole function can have an arbitrarily large number of connected components. Many of the surfaces we construct have trivial automorphism group, and are the first examples of local maxima with this property.