McShane's identity, using elliptic elements

作者：Schmidt Thomas A^*; Sheingorn Mark

来源：Geometriae Dedicata, 2008, 134(1): 75-90.

DOI：10.1007/s10711-008-9246-0

• McShane's identity
• hyperbolic surface
• geodesic length

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.

• 出版日期2008-6