Geometriae Dedicata

, Volume 134, Issue 1, pp 75–90 | Cite as

McShane’s identity, using elliptic elements

Original Paper


We introduce a new method to establish McShane’s Identity. Elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics. The highest points of simple closed geodesics are already given as the appropriate complement of the regions excluded by those elements of order two that factor hyperbolic elements whose axis projects to be simple. The widths of the intersection with an appropriate horocycle of the excluded regions sum to give McShane’s value of 1/2. The remaining points on the horocycle are highest points of simple open geodesics, we show that this set has zero Hausdorff dimension.


McShane’s identity Hyperbolic surface Geodesic length 

Mathematics Subject Classification (2000)

57M50 20H10 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Oregon State UniversityCorvallisUSA
  2. 2.CUNY - Baruch CollegeNew YorkUSA

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