Low height geodesics and the Markoff spectrum

2004/10/07 by Thomas A. Schmidt, Schmidt, Thomas A., Mark Sheingorn +1
Mathematics · #11J70 #30F35 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0410197

openalex publication_date 2004/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify, in terms of topology of highest arcs, low height non-simple geodesics on the modular hyperbolic punctured sphere with three elliptic fixed points of order two. Of eight possible types, exactly one consists of geodesics that form a bigon about the cusp; we express all such geodesics in terms of Markoff triples. We also show that any geodesic joining two elliptic fixed points has locally isolated height.

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