2005/01/20 by Martin Möller, Moeller, Martin · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/0501333
openalex publication_date 2005/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura curves and of Teichmueller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces. Finally we translate our main result into a classification of Teichmueller curves with totally degenerate Lyapunov spectrum.