Simple geodesics and a series constant over Teichmüller space (Q1127756)

Let \(M\) be a finite area Riemann surface with one cusp and no ramification. This paper offers the ramification following the spectacular identity: \(\sum{1\over 1+ \exp {1\over 2} (|\alpha |+ |\beta |)}= {1\over 2}\), where the sum is over all pairs of simple closed geodesics \(\alpha,\beta\) which bound an embedded pair of pants containing the cusp and \(|\;|\) gives the length. Additionally, the paper characterizes the \textit{Birman Series set} BS (the set of all points on all simple geodesics on \(M\) (which now may have more than one cusp)) near a fixed cusp. \textit{Near} means a horocyclic region \(H\) small enough so that any simple geodesic entering it must not leave; must terminate \textit{at} the cusp. The characterization is a bit too involved to state here, but it is given in terms of the isolated and boundary points of \(BS\cap \partial H\); which are themselves specified succintly and geometrically.