Generalized Markoff maps and McShane's identity |
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Authors: | Ser Peow Tan Yan Loi Wong |
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Institution: | a Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore b School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, People's Republic of China |
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Abstract: | We study the (relative) SL(2,C) character varieties of the one-holed torus and the action of the mapping class group on the (relative) character variety. We show that the subset of characters satisfying two simple conditions called the Bowditch Q-conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset. Furthermore, this is the largest open subset for which this holds. We also show that a generalization of McShane's identity holds for all characters satisfying the Bowditch Q-conditions. Finally, we show that further variations of the McShane-Bowditch identity hold for characters which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch Q-conditions, with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle, and also for closed hyperbolic 3-manifolds which are obtained by hyperbolic Dehn surgery on such manifolds. |
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Keywords: | One-holed torus Character variety Mapping class group McShane's identity Punctured torus bundle Hyperbolic Dehn surgery |
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