McShane's identity explained

T with a complete, finite-volume hyperbolic structure is given by\sum _{\gamma} (1)/(= 1+ e ^{\ell(\gamma)}) (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]

Further reading

• Tan . Ser Peow . Wong . Yan Loi . Zhang . Ying . Necessary and Sufficient Conditions for Mcshane's Identity and Variations . Geometriae Dedicata . April 2006 . 119 . 1 . 199–217 . 10.1007/s10711-006-9069-9 . math/0411184 . 17575980 .
• McShane . Greg . Simple geodesics and a series constant over Teichmuller space . Inventiones Mathematicae . 8 May 1998 . 132 . 3 . 607–632 . 10.1007/s002220050235 . 16362716 .

Notes and References

1. . Mirzakhani . Maryam . 2004 . Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves .