Generalized Markoff Maps and McShane's Identity

Abstract

We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations and that the subset of representations satisfying these conditions is the largest open subset in the relative character variety on which the mapping class group acts properly discontinuously. Moreover we generalize Bowditch's results on variations of McShane's identity for complete, finite volume hyperbolic 3-manifolds which fiber over the circle, with the fiber a punctured-torus, to identities for incomplete hyperbolic structures on such manifolds, hence obtaining identities for closed hyperbolic 3-manifolds which are obtained by doing hyperbolic Dehn surgery on such manifolds.

49 pages, 9 figures

References in corpus (2)

• The Modular Group Action on Real SL(2)-characters of a One-Holed Torus
• The SL(2,C) character variety of the one-holed torus

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• The realization problem for Jørgensen numbers
• The Tri-Pants Graph of the Twice-Punctured Torus
• Primitive stability and the Bowditch conditions revisited
• McShane's Identity in Rank One Symmetric Spaces
• Dynamics on the SU(2,1)-character variety of the one-holed torus
• A survey of length series identities for surfaces, % 3-manifolds and representation varieties