Teichmuller theory of the punctured solenoid

The punctured solenoid H plays the role of an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmuller space of H is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of H. Furthermore, a point in the decorated Teichmuller space induces a polygonal decomposition of H itself giving a combinatorial description of the decorated Teichmuller space. This is used to obtain a non-trivial set of generators of the modular group of H, and each word in these generators admits a normal form. There is furthermore a non-degenerate modular group invariant two form on the Teichmuller space of H. All of this structure is in perfect analogy with that of the decorated Teichmuller space of a punctured surface of finite type.