Teichmüller Theory of the Punctured Solenoid

The punctured solenoid H is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller 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 Teichmüller space induces a polygonal decomposition of H giving a combinatorial description of its decorated Teichmüller space itself. This is used to obtain a non-trivial set of generators of the modular group of H, which is presumably the main result of this paper. Moreover, each word in these generators admits a normal form, and the natural equivalence relation on normal forms is described. There is furthermore a non-degenerate modular group invariant two form on the Teichmüller space of H. All of this structure is in perfect analogy with that of the decorated Teichmüller space of a punctured surface of finite type. Stony Brook IMS Preprint #2005/06 August 2005