A survey of length series identities for surfaces, % 3-manifolds and representation varieties
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We survey some of our recent results on length series identities for hyperbolic (cone) surfaces, possibly with cusps and/or boundary geodesics; classical Schottky groups; representations/characters of the one-holed torus group to $SL(2, \mathbf C)$; and hyperbolic 3 manifolds obtained by hyperbolic Dehn surgery on punctured torus bundles over the circle. These can be regarded as generalizations and variations of McShane's identity for cusped hyperbolic surfaces, which has found some striking applications in the recent work of Mirzakhani. We discuss some of the methods and techniques used to obtain these identities.
22 pages, submitted to proceedings of symposium "Complex Analysis and Geometry of Hyperbolic Spaces", Dec 2005, RIMS, Kyoto, Japan
22 pages, submitted to proceedings of symposium "Complex Analysis and Geometry of Hyperbolic Spaces", Dec 2005, RIMS, Kyoto, Japan