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McShane's identity

From Wikipedia, the free encyclopedia

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
  • (γ) denotes the hyperbolic length of γ.

This identity was generalized by Maryam Mirzakhani in her PhD thesis[1]

References

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

Further reading

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