The SL(2,C) character variety of the one-holed torus

Ser Peow Tan; Yan Loi Wong; Ying Zhang

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The SL(2,C) character variety of the one-holed torus

Ser Peow Tan·Yan Loi Wong·Ying Zhang

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Abstract

In this note we announce several results concerning the SL(2,C) character variety ${\mathcal X}$ of the one-holed torus. We give a description of the largest open subset ${\mathcal X}_{BQ}$ of ${\mathcal X}$ on which the mapping class group $Γ$ acts properly discontinuously, in terms of two very simple conditions, and show that a series identity generalizing McShane's identity for the punctured torus holds for all characters in this subset. We also give variations of the McShane-Bowditch identities to characters fixed by an Anosov element of $Γ$ with applications to closed hyperbolic three manifolds. Finally we give a definition of end invariants for SL(2,C) characters and give a partial classification of the set of end invariants of a character in ${\mathcal X}$.

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Ser Peow Tan, Yan Loi Wong, Ying Zhang. 2005-09-02. The SL(2,C) character variety of the one-holed torus. https://arxiv.org/abs/math/0509033

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