McShane's identity
{{Morefootnotes|article|date=May 2023}}In geometric topology, McShane's identity for a once punctured torus with a complete, finite-volume hyperbolic structure is given by
:
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{{cite thesis |id={{ProQuest|305191605}} |last1=Mirzakhani |first1=Maryam |date=2004 |title=Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves }}
References
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Further reading
- {{cite journal |last1=Tan |first1=Ser Peow |last2=Wong |first2=Yan Loi |last3=Zhang |first3=Ying |title=Necessary and Sufficient Conditions for Mcshane's Identity and Variations |journal=Geometriae Dedicata |date=April 2006 |volume=119 |issue=1 |pages=199–217 |doi=10.1007/s10711-006-9069-9 |arxiv=math/0411184 |s2cid=17575980 }}
- {{cite journal |last1=McShane |first1=Greg |title=Simple geodesics and a series constant over Teichmuller space |journal=Inventiones Mathematicae |date=8 May 1998 |volume=132 |issue=3 |pages=607–632 |doi=10.1007/s002220050235 |s2cid=16362716 }}