McShane's identity

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In geometric topology, McShane's identity for a once punctured torus \mathbb{T} with a complete, finite-volume hyperbolic structure is given by

\sum_\gamma \frac{1}{1 + e^{\ell(\gamma)}}=\frac{1}{2}

where

  • the sum is over all simple closed geodesics γ on the torus; and
  • (γ) denotes the hyperbolic length of γ.

References[edit]

  • 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.