Categoricity and non-arithmetic Fuchsian groups

TL;DR

This paper establishes a model-theoretic framework for non-arithmetic Fuchsian groups, showing their associated theories are categorical, complete, and stable, extending previous results from the arithmetic case.

Contribution

Introduces a natural $L_{_1,}$-axiomatization for the theory of $j_{_{ ext{Gamma}}}$ as a covering map, proving categoricity and stability results in the non-arithmetic setting.

Findings

The $L_{_1,}$-theory $T^{}_{SF}$ is categorical in all infinite cardinalities.

The first-order theory $T_{j_{_{ ext{Gamma}}}}$ is complete and admits quantifier elimination.

The theory $T_{j_{_{ ext{Gamma}}}}$ is $$-stable.

Abstract

Let $\Gamma \subset PSL_2(\mathbb{R})$ be a non-arithmetic Fuchsian group of the first kind with finite covolume, and let $j_{\Gamma}$ be a corresponding uniformizer. In this paper we introduce a natural $L_{\omega_1,\omega}$-axiomatization $T^{\infty}_{SF}$ of the theory of $j_{\Gamma}$ 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 $T_{j_{\Gamma}}$ is complete, admits elimination of quantifiers, and is $\omega$-stable.