A systolic geometric cell decomposition for the space of once-holed Riemann surfaces of genus 2 (Q5954111)

Let \(S_L\) be the set of all Riemann surfaces \((M,g)\) of genus \(2\) which have one boundary geodesic of length \(2L\). Suppose \((M,g)\) has the longest systole among elements of \(S_L\). The author shows that \((M,g)\) is isometric to one of three surfaces which are constructed explicitly and which have exactly nine systoles. This result is closely related to a major problem in the hyperbolic geometry of numbers -- the problem of finding the closed Riemann surface of genus 3 with the longest systole.