  In this paper, we prove several Ax-Schanuel type results for uniformizers of
geometric structures. In particular, we give a proof of the full Ax-Schanuel
Theorem with derivatives for uniformizers of any Fuchsian group of the first
kind and any genus. Our techniques combine tools from differential geometry,
differential algebra and the model theory of differentially closed fields. The
proof is very similar in spirit to Ax's proof of the theorem in the case of the
exponential function.
