Teichmueller curves, triangle groups, and Lyapunov exponents

TL;DR

This paper constructs specific Teichmueller curves associated with triangle groups, explores their properties using hypergeometric operators, and calculates their Lyapunov exponents, extending previous work by Veech and Ward.

Contribution

It introduces a new construction of Teichmueller curves uniformized by (m,n,∞) triangle groups, generalizing prior examples and linking Lyapunov exponents to line bundle degrees.

Findings

Constructed Teichmueller curves for all m<n using hypergeometric operators.

Identified Billiard tables generating these curves for small m.

Calculated Lyapunov exponents for the constructed Teichmueller curves.

Abstract

We construct a Teichmueller curve uniformized by the Fuchsian triangle group (m,n,\infty) for every m<n. Our construction includes the Teichmueller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find Billiard tables that generate these Teichmueller curves. We interprete some of the so-called Lyapunov exponents of the Kontsevich--Zorich cocycle as normalized degrees of some natural line bundles on a Teichmueller curves. We determine the Lyapunov exponents for the Teichmueller curves we construct.