Necessary and sufficient conditions for McShane's identity and variations

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dc.creatorTan, Ser Peow
dc.creatorWong, Yan Loi
dc.creatorZhang, Ying
dc.date2004-11-09
dc.date2005-01-11
dc.date.accessioned2026-07-07T07:37:26Z
dc.date.available2026-07-07T07:37:26Z
dc.descriptionGreg 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 Γto SL(2,C). In this note we extend Bowditch's result by giving necessary and sufficient conditions for the identity to hold, and also for the generalized McShane identity to hold for arbitrary (not necessarily type preserving) representations. We also give a version of Bowditch's variation of McShane's identity to once-punctured torus bundles, to the case where the monodromy is generated by a reducible element, and provide necessary and sufficient conditions for the variation to hold.
dc.descriptionMinor corrections, acknowlegement and correct reference to Akiyoshi-Miyachi-Sakuma for Theorem 1.3, and finally, a new result (Theorem 2.6) added
dc.identifierhttps://arxiv.org/abs/math/0411184
dc.identifierhttp://arxiv.org/abs/math/0411184
dc.identifierGeometriae Dedicata, 119 (2006), no. 1, 199--217
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/120777
dc.subjectGeometric Topology
dc.subjectDifferential Geometry
dc.subject32G15;57M50;30F60
dc.titleNecessary and sufficient conditions for McShane's identity and variations
dc.typetext

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