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Teichmuller theory of the punctured solenoid

R. C. Penner·Dragomir Saric
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Abstract

The punctured solenoid $§$ 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 $§$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $§$. Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of $§$ 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 $§$, 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 $§$. All of this structure is in perfect analogy with that of the decorated Teichmüller space of a punctured surface of finite type.

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R. C. Penner, Dragomir Saric. 2005-08-24. Teichmuller theory of the punctured solenoid. https://arxiv.org/abs/math/0508476

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