McShane's identity
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In geometric topology, McShane's identity for a once punctured torus [math]\displaystyle{ \mathbb{T} }[/math] with a complete, finite-volume hyperbolic structure is given by
- [math]\displaystyle{ \sum_\gamma \frac{1}{1 + e^{\ell(\gamma)}}=\frac{1}{2} }[/math]
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[1]
References
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.
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