Construction of hyperbolic Riemann surfaces with large systoles

Abstract

Let S be a compact hyperbolic Riemann surface of genus ${g \geq 2}$. We call a systole a shortest simple closed geodesic in S and denote by ${{\rm sys}(S)}$ its length. Let ${{\rm msys}(g)}$ be the maximal value that ${{\rm sys}(\cdot)}$ can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface S _max a compact Riemann surface of genus g whose systole has length ${{\rm msys}(g)}$. In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating ${{\rm msys}(\cdot)}$ of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.