Categoricity and non-arithmetic Fuchsian groups
John Baldwin, Joel Nagloo
TL;DR
The paper proves that for a non-arithmetic Fuchsian group $\Gamma$ of the first kind with finite covolume, the infinitary theory $T^{\infty}_{SF}$ of the uniformizing map $j_{\Gamma}$ is categorical in all infinite cardinals, while the associated first-order theory $T_{j_{\Gamma}}$ is complete, eliminates quantifiers, and is $\omega$-stable. The authors develop a two-sorted $L_{\omega_1,\omega}$-axiomatization that encodes the covering structure, establish a definable set $Z_{\mathbf{g}}$ via Gamma-special polynomials, and obtain a back-and-forth that yields $\infty$-$\omega$-equivalence of models. A key technical simplification in the non-arithmetic case is used to relate $\tau$-types to field-types, enabling a direct reduction to the base field, together with a robust back-and-forth framework. Finally, the class forms an almost quasiminimal excellent geometry, leading to categoricity in all uncountable powers and clarifying the interaction between hyperbolic geometry and model-theoretic stability. This extends previous arithmetic results and demonstrates categoricity independent of arithmeticity, with potential implications for higher-dimensional uniformizations and related Shimura-type contexts.
Abstract
Let $Γ\subset PSL_2(\mathbb{R})$ be a non-arithmetic Fuchsian group of the first kind with finite covolume, and let $j_Γ$ be a corresponding uniformizer. In this paper we introduce a natural $L_{ω_1,ω}$-axiomatization $T^{\infty}_{SF}$ of the theory of $j_Γ$ viewed as a covering map. We show that $T^{\infty}_{SF}$ is categorical in all infinite cardinalities, extending to the non-arithmetic setting earlier results of Daw and Harris obtained in the arithmetic case. We also show that the associated first-order theory $T_{j_Γ}$ is complete, admits elimination of quantifiers, and is $ω$-stable.