A systolic geometric cell decomposition for the space of once-holed Riemann surfaces of genus 2

Let ξ⩾0 be real. We show that the Riemann surface, of genus 2 with one boundary geodesic of length 2ζ, with the longest systole is isometric to one of three surfaces. These three surfaces are explicitly constructed and they all have exactly nine systoles. This result “almost” solves a major problem in the hyperbolic geometry of numbers, namely, the problem of finding the closed Riemann surface of genus 3 with the longest systole.