Construction of hyperbolic Riemann surfaces with large systoles

Abstract

Let $S$ be a compact hyperbolic Riemann surface of genus $g\ge 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathrm{sys}(S)$ its length. Let $msys(g)$ be the maximal value that $\mathrm{sys}(\cdot )$ 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 $\mathrm{msys}(g)$. 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 $\mathrm{msys}(\cdot )$ of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.

17 pages, 2 figures, final version, Journal of Geometry

References in corpus (2)

• Systoles and kissing numbers of finite area hyperbolic surfaces
• A Note on Riemann Surfaces of Large Systole