Shimura- and Teichmueller curves

TL;DR

This paper classifies curves in the moduli space that are both Shimura and Teichmueller, revealing only one such curve exists beyond genus one, using Hodge theory and geometric analysis.

Contribution

It provides the first classification of Shimura- and Teichmueller curves, connecting Hodge theory, geometry of square-tiled coverings, and Lyapunov spectrum analysis.

Findings

Only one Shimura- and Teichmueller curve exists beyond genus one.

The classification is achieved through Hodge-theoretic and geometric methods.

Teichmueller curves with totally degenerate Lyapunov spectrum are classified.

Abstract

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.