Schwarz triangle mappings and Teichm\"uller curves: abelian square-tiled surfaces

TL;DR

This paper explores the geometry of Teichm"uller curves generated by abelian square-tiled surfaces, linking period mappings to Schwarz triangle mappings and computing Lyapunov exponents explicitly.

Contribution

It introduces a geometric description of period mappings via Schwarz triangle mappings for certain Teichm"uller curves and provides explicit formulas for Lyapunov exponents.

Findings

Lyapunov exponents are ratios of areas of hyperbolic triangles.

Period mappings can be described using Schwarz triangle mappings.

Explicit computation of Lyapunov exponents for abelian square-tiled surfaces.

Abstract

We consider normal covers of CP^1 with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichm\"uller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichm\"uller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov exponents in terms of the period mapping. In the case of abelian square-tiled surfaces, the Lyapunov exponents are ratios of areas of hyperbolic triangles.