Cocompact Fuchsian groups with a modular embedding

2025/03/16 by Stover, Matthew
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.12656

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: \mathbbB¹ → \mathbbB¹ intertwining the actions of Γ and Γ^σ on the Poincaré disk \mathbbB¹. 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 that there are infinitely many signatures for which no finite-volume quotient of \mathbbB¹ with that signature can be an immersed totally geodesic curve on a complex hyperbolic 2-orbifold.

Related

• Modular embeddings and automorphic Higgs bundles
• Representation growth of Fuchsian groups and modular forms
• Many Examples of Non-cocompact Fuchsian Groups Sitting in PSL(2,Q)
• Action of ℝ-Fuchsian groups on ℙ_ℂⁿ
• A classification of \mathbb C-Fuchsian subgroups of Picard modular groups