Construction of hyperbolic Riemann surfaces with large systoles

2013/05/23 by Hugo Akrout, Akrout, Hugo, Bjoern Muetzel +1
Mathematics · #30F10 #32G15 #53C22 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1305.5510

openalex publication_date 2013/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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