ar X iv : m at h / 05 02 46 4 v 1 [ m at h . G T ] 2 2 Fe b 20 05 GENERALIZED MARKOFF MAPS AND MCSHANE ’ S IDENTITY

Following Bowditch [5], we study representations of the free group on two generators into SL(2, C), and the connection with generalized Markoff maps. We show that Bowditch’s Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations. These conditions are very close to being necessary as well, and a large class of representations arising from important and interesting geometric constructions satisfy these conditions. We also show that the subset of representations satisfying these conditions is open in the relative character variety, and it is the largest open subset 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.