Cocompact Fuchsian groups with a modular embedding

Abstract

A Fuchsian group $\hat{I}“$ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate $\hat{I}{“}^{\ddot{I}}ƒ$ comes equipped with a holomorphic (or conjugate holomorphic) map $\ddot{I}{•}^{\ddot{I}}ƒ:{\mathbb{B}}^1\to {\mathbb{B}}^1$ intertwining the actions of $\hat{I}“$ and $\hat{I}{“}^{\ddot{I}}ƒ$ on the Poincaré disk ${\mathbb{B}}^1$. This paper provides the first cocompact nonarithmetic Fuchsian groups with a modular embedding that are not commensurable with a triangle group. The main result, proved using period domains, is that any immersed totally geodesic complex curve on a complex hyperbolic $2$-orbifold has a modular embedding. Another consequence is arithmeticity of totally geodesic curves on finite-volume complex hyperbolic surfaces that are commensurable with quotients of ${\mathbb{B}}^1$ by the group generated by reflections in quadrilaterals satisfying certain angle conditions.

To appear in IMRN