Cocompact Fuchsian groups with a modular embedding

TL;DR

The paper addresses the existence and structure of cocompact nonarithmetic Fuchsian groups with modular embeddings that are not commensurable with triangle groups. It employs period-domain and Mumford–Tate domain techniques to show that immersed totally geodesic curves in finite-volume complex hyperbolic $2$-orbifolds admit modular embeddings, linking Galois conjugates to holomorphic intertwining maps. The main contributions include (i) the construction of the first compact nonarithmetic examples with modular embeddings not arising from triangle groups, (ii) a general theorem that any immersed totally geodesic complex curve in a finite-volume complex hyperbolic $2$-orbifold has a modular embedding, and (iii) consequences for the arithmeticity of certain reflection groups associated to quadrilateral reflection groups. Together, these results provide a cocompact counterpoint to prior noncompact examples and yield infinitely many semi-arithmetic compact quotients that admit no proper totally geodesic immersion, with implications for Hilbert modular varieties and related period-map phenomena.

Abstract

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 ${φ^σ: \mathbb{B}^1 \to \mathbb{B}^1}$ intertwining the actions of $Γ$ and $Γ^σ$ 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.

Figures (1)

• Figure 1: Quadrilaterals with vertices at the fixed points of $a$, $b$, $c$, and $abc$

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Remark 2.1
• Lemma 2.2
• Corollary 2.3
• proof
• Proposition 3.1
• proof