Teichmuller theory of the punctured solenoid

Abstract

The punctured solenoid $\S$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichm\"uller space of $\S$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $\S$. Furthermore, a point in the decorated Teichm\"uller space induces a polygonal decomposition of $\S$ giving a combinatorial description of its decorated Teichm\"uller space itself. This is used to obtain a non-trivial set of generators of the modular group of $\S$, 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 Teichm\"uller space of $\S$. All of this structure is in perfect analogy with that of the decorated Teichm\"uller space of a punctured surface of finite type.