A series associated to generating pairs of a once punctured torus group and a proof of McShane's identity
In his paper [Bull. Lond. Math. Soc. 36, No. 2, 181--187 (2004; Zbl 1052.57021)], \textit{G. McShane} proved that for any finite-area hyperbolic metric on a punctured torus, one has the following identity: \[ \sum_{\gamma}\arcsin \left(\frac{1}{\cosh(|\gamma|/2)}\right)=\pi/2 \] where the sum is taken over all simple closed geodesics passing through a fixed pair of Weierstrass points. In the paper under review, the author provides a new proof of this identity based on an investigation of the arrangement of axes of simple hyperbolic elements in a once-punctured torus-group which are represented by palindromic words. The author's argument also includes a short proof of the fact that the linear measure of the infinitesimal Birman-Series set is zero.
scientific article; zbMATH DE number 5953595