McShane's identity
# 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.

This identity was generalized by Maryam Mirzakhani on her PhD thesis

