McShane's identity

Template:MorefootnotesIn geometric topology, McShane's identity for a once punctured torus $𝕋$ 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 (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

Further reading

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• Template:Cite journal