Ahlfors Bers Colloquium
May
19-22, 2005
University of
Michigan
Ann Arbor, Michigan
Registration
Organizers
Dick
Canary
Juha
Heinonen
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Workshop
Abstract Detail
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Dragomir Saric Stony Brook
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Decorated Teichm\" uller space of the punctured solenoid
We present a joint work with R. Penner. The punctured solenoid $\mathcal{S}$ is the inverse limit of the directed system of all finite sheeted unbranched coverings of the punctured torus. The solenoid $\mathcal{S}$ supports many hyperbolic metrics $\mathcal{S}_m$. We define the decorated Teichm\" uller space $\tilde{T}(\mathcal{S})$ of the punctured solenoid. A decoration $\tilde{\mathcal{S}}_m$ on the (hyperbolic) solenoid $\mathcal{S}_m$ is an assignment of a horocycle to each puncture. The decorations $\mathcal{S}_m\in \tilde{T}(\mathcal{S})$ produce lambda lengths which parametrize $\tilde{T}(\mathcal{S})$. Moreover, a decoration $\tilde{\mathcal{S}}_m$ ``almost always'' induces an ideal triangulation of $\mathcal{S}_m$. This gives a combinatorial ``decomposition'' on $\tilde{T}(\mathcal{S})$ and we use this picture to give a presentation of the modular group of $\mathcal{S}$. We also discuss a generalization of the Weil-Petersson two form on $\tilde{T}(\mathcal{S})$ and its push-forward to the Teichm\" uller space of $\mathcal{S}$. | ||
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