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Title: Necessary and sufficient conditions for Mcshane's identity and variations
Authors: Tan, S.P. 
Wong, Y.L. 
Zhang, Y.
Keywords: Character variety
Farey tessellation
McShane's identity
Once punctured torus bundle
Representation variety
Issue Date: Apr-2006
Citation: Tan, S.P., Wong, Y.L., Zhang, Y. (2006-04). Necessary and sufficient conditions for Mcshane's identity and variations. Geometriae Dedicata 119 (1) : 199-217. ScholarBank@NUS Repository.
Abstract: Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two generators Γ to SL(2,C), this was further generalized by the authors to obtain sufficient conditions for a generalized McShane's identity to hold for arbitrary (not necessarily type-preserving) non-reducible representations in Tan et al. (Submitted). Here we extend the above by giving necessary and sufficient conditions for the generalized McShane identity to hold (Akiyoshi, Miyachi and Sakuma had proved it for type-preserving representations). We also give a version of Bowditch's variation of McShane's identity to once-punctured torus bundles, in the case where the monodromy is generated by a reducible element, and provide necessary and sufficient conditions for the variations to hold.
Source Title: Geometriae Dedicata
ISSN: 00465755
Appears in Collections:Staff Publications

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