McShane's identity, using elliptic elements (Q930527)

The paper provides a new proof of McShane's identity: \(2\sum_\gamma (1+e^{\ell(\gamma)})^{-1}=1\), where the sum is over all simple closed geodesics of any hyperbolic once-punctured torus and \(\ell(\gamma)\) denotes the (hyperbolic) length of \(\gamma\). The strategy of the proof is to consider simple geodesics in the hyperbolic once-punctured sphere with three cone points of order \(2\): there is a natural one-to-one correspondence between such spheres and hyperbolic once-punctured tori realized by considering the quotient of a torus by its Weierstrass involution. Simple closed geodesics in the once-punctured torus map to simple closed geodesics in the once-punctured sphere (these latter geodesics correspond to arcs between cone points) and all simple closed geodesics in the once-punctured sphere are obtained this way. A key observation is then to notice that hyperbolic elements of the Fuchsian group which uniformizes a once-punctured sphere with three cone points, whose axes map to simple closed geodesics, are products of elliptic elements (of order \(2\)).