McShane's identity

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
• 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
• McShane, G. Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998), no. 3, 607–632.