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Title: Generalized Markoff maps and McShane's identity
Authors: Tan, S.P. 
Wong, Y.L. 
Zhang, Y.
Keywords: Character variety
Hyperbolic Dehn surgery
Mapping class group
McShane's identity
One-holed torus
Punctured torus bundle
Issue Date: 30-Jan-2008
Source: Tan, S.P., Wong, Y.L., Zhang, Y. (2008-01-30). Generalized Markoff maps and McShane's identity. Advances in Mathematics 217 (2) : 761-813. ScholarBank@NUS Repository.
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. © 2007 Elsevier Inc. All rights reserved.
Source Title: Advances in Mathematics
ISSN: 00018708
DOI: 10.1016/j.aim.2007.09.004
Appears in Collections:Staff Publications

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