A survey of length series identities for surfaces, % 3-manifolds and representation varieties

Details A survey of length series identities for surfaces, % 3-manifolds and representation varieties A survey of length series identities for surfaces, % 3-manifolds and representation varieties

2006-07-05

arxiv.org/abs/math/0607107v1

RIMS Kokyuroku 1518, October 2006, pp. 20-41.

Mathematics

Geometric Topology

22 pages, submitted to proceedings of symposium "Complex Analysis and Geometry of Hyperbolic Spaces", Dec 2005, RIMS, Kyoto, J

Scientific paper

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.

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