Simple Geodesics and a Series Constant over Teichmuller Space

We investigate the Birman Series set in a neighborhood of a cusp on a punctured surface, showing that it is homeomorpic to a cantor set union countably many isolated points cross a line. The local topology of the cantor set is shown to be related in a simple way to the global behavior of simple geodesics. From this we deduce that a certain series is constant across the Teichmuller space 1. Introduction In 10], by analysing the pointset of all simple geodesics on the torus, the following striking identity for the lengths of closed simple geodesics was obtained: