Categoricity and non-arithmetic Fuchsian groups

Abstract

Let $\hat{I}“\subset PS{L}_2(\mathbb{R})$ be a non-arithmetic Fuchsian group of the first kind with finite covolume, and let $j_{\hat{I}}“$ be a corresponding uniformizer. In this paper we introduce a natural $L_{\ddot{I}{‰}_1,\ddot{I}‰}$-axiomatization $T_{SF}^{\infty }$ of the theory of $j_{\hat{I}}“$ viewed as a covering map. We show that $T_{SF}^{\infty }$ 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 $T_{j_{\hat{I}}“}$ is complete, admits elimination of quantifiers, and is $\ddot{I}‰$-stable.