  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.
