McShane's identity

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

${\displaystyle \sum _{\gamma }{\frac {1}{1+e^{\ell (\gamma )}}}={\frac {1}{2}}}$

where

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

This identity was generalized by Maryam Mirzakhani in her PhD thesis^[1]