Cocompact Fuchsian Groups With a Modular Embedding

A Fuchsian group Γ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate Γ ^σ comes equipped with a holomorphic (or conjugate holomorphic) map φ ^σ : B¹ → B¹ intertwining the actions of Γ and Γ ^σ on the Poincaré disk B¹. 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 B¹ by the group generated by reflections in quadrilaterals satisfying certain angle conditions.