Cocompact Fuchsian Groups With a Modular Embedding

Abstract A Fuchsian group $\Gamma $ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate $\Gamma ^\sigma $ comes equipped with a holomorphic (or conjugate holomorphic) map ${\phi ^\sigma : \mathbb{B}^{1} \to \mathbb{B}^{1}}$ intertwining the actions of $\Gamma $ and $\Gamma ^\sigma $ 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.