McShane's identity

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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 (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]

References

^ Mirzakhani, Maryam (2004). Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves (Thesis). ProQuest 305191605.

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. arXiv:math/0411184. doi:10.1007/s10711-006-9069-9. S2CID 17575980.
McShane, Greg (8 May 1998). "Simple geodesics and a series constant over Teichmuller space". Inventiones Mathematicae. 132 (3): 607–632. doi:10.1007/s002220050235. S2CID 16362716.

[1] Mirzakhani, Maryam (2004). Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves (Thesis). ProQuest 305191605.

[1]

References

Further reading