Generalized Markoff maps and McShane's identity
By S. Tan, Yan Loi Wong, Ying Zhang. 2005
60 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/4a2f9c134e26e38fc16164d2ec0623e3989b755d
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- MAPPING CLASS GROUP DYNAMICS ON SURFACE GROUP REPRESENTATIONS (2005, background): “Tan, Wong and Zhang [ 97 ] have shown that the action of Mod� on the representations satisfying the analogue of Bowditch’s Q-conditions is proper.”