Discoveries / geometry-topology

Large systoles in every sufficiently large genus

The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant $2/9$ to $1$. The every-genus ladder it climbs is Katz-Sabourau's $19/120$ and then Liu-Petri's $2/9$, the latter also by a random construction. Constant $1$ was already reached by Petri-Walker along a subsequence of genera, following Erdos-Sachs, so the new contribution is achieving it uniformly rather than the constant itself. The asymptotic problem stays open, and the remaining gap is wide: Brooks and Buser-Sarnak give $\limsup\ge4/3$, while the elementary area bound is $\max\mathrm{sys}(S)\le2\log(4g-2)$, asymptotically $2\log g$. So this closes much of the liminf gap and determines no optimal constant.