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

Further reading

• Tan, Ser Peow; Wong, Yan Loi; Zhang, Ying (April 2006). "Necessary and Sufficient Conditions for Mcshane's Identity and Variations". Geometriae Dedicata 119 (1): 199–217. doi:10.1007/s10711-006-9069-9. 
• McShane, Greg (8 May 1998). "Simple geodesics and a series constant over Teichmuller space". Inventiones Mathematicae 132 (3): 607–632. doi:10.1007/s002220050235.