## Institute for Mathematical Sciences

## Preprint ims05-06

** R. C. Penner and Dragomir Saric**
* 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.

View ims05-06 (PDF format)