Shimura- and Teichmueller curves
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.
Keywords
Cite
@article{arxiv.math/0501333,
title = {Shimura- and Teichmueller curves},
author = {Martin Moeller},
journal= {arXiv preprint arXiv:math/0501333},
year = {2010}
}
Comments
Major revision. Main result weakened concerning the case of genus 5 curves. One missing case added in genus 4. Application to degenerate Lyapunov spectra added