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In [[geometric topology]], '''McShane's identity''' for a once punctured [[torus]] <math>\mathbb{T}</math> with a complete, finite volume hyperbolic structure is given by
 
:<math>\sum_\gamma \frac{1}{1 + e^{l(\gamma)}}=\frac{1}{2}</math>
 
The sum is over all simple closed [[geodesics]] γ on the torus; here ''l''(γ) denotes the hyperbolic length of γ.
 
==References==
*''Necessary and Sufficient Conditions for McShane's Identity and Variations'' Ser Peow Tan, Yan Loi Wong, and Ying Zhang eprint arXiv:math/0411184 [http://arxiv.org/abs/math/0411184v2]
*McShane, G. Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998), no. 3, 607–632.
 
[[Category:Topology]]
[[Category:Geometric topology]]

Latest revision as of 14:55, 25 January 2014

In geometric topology, McShane's identity for a once punctured torus with a complete, finite volume hyperbolic structure is given by

The sum is over all simple closed geodesics γ on the torus; here l(γ) denotes the hyperbolic length of γ.

References

  • Necessary and Sufficient Conditions for McShane's Identity and Variations Ser Peow Tan, Yan Loi Wong, and Ying Zhang eprint arXiv:math/0411184 [1]
  • McShane, G. Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998), no. 3, 607–632.

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