Categoricity and non-arithmetic Fuchsian groups

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.

💡 Research Summary

The paper studies the model‑theoretic properties of the uniformizing function j_Γ associated with a non‑arithmetic Fuchsian group Γ (of the first kind and finite covolume). The authors introduce a natural infinitary axiomatization T^{∞}{SF} in the language L{ω₁,ω} that captures the two‑sorted structure consisting of the covering domain D (and the action of the commensurator G = Comm(Γ) on it) and the target algebraic curve S together with the covering map J: D → S.

The main results are twofold. Theorem A shows that every model of T^{∞}{SF} is (∞, ω)‑equivalent, i.e., all models of any infinite cardinality satisfy the same Scott sentence. Consequently, the associated first‑order theory T{j_Γ} is complete, admits quantifier elimination, and is ω‑stable. The proof relies on a careful analysis of special points (fixed points of elements of G) and Hodge‑generic points, the introduction of constant symbols for the images of special points, and a “determined type” argument showing that the type of any element is fixed by the Γ‑special polynomials Φ_g that encode Hecke correspondences.

Theorem B establishes full categoricity of T^{∞}_{SF} in all infinite cardinalities. While the arithmetic case (treated by Daw and Harris) requires deep open‑image theorems for Galois representations attached to Hecke orbits, the non‑arithmetic setting avoids these heavy tools. Instead, the authors use the fact that