We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painleve irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem which have interesting consequences. In particular, we apply our results to answer a question of Painleve (1895). We also answer certain cases of the Andre-Pink conjecture, namely in the case of orbits of commensurators of Fuchsian groups.

中文翻译：

---

Ax-Lindemann-Weierstrass 与衍生物和属 0 Fuchsian 群

我们证明了 Ax-Lindemann-Weierstrass 定理和第一类零属 Fuchsian 群的均匀化函数的导数。我们的证明依赖于微分伽罗瓦理论、线性微分方程的单调性、对代数和 Liouvillian 解的研究、西冈对某些 Schwarzian 方程的 Painleve 不可约性的微分代数工作，以及来自微分封闭域模型理论的大量机制。我们的技术允许对 Ax-Lindemann-Weierstrass 定理进行某些推广，从而产生有趣的结果。特别是，我们应用我们的结果来回答 Painleve (1895) 的一个问题。我们还回答了安德烈-平克猜想的某些情况，即 Fuchsian 群的公合子轨道的情况。