Low height geodesics and the Markoff spectrum
  We classify, in terms of topology of highest arcs, low height non-simple geodesics on the modular hyperbolic punctured sphere with three elliptic fixed points of order two. Of eight possible types, exactly one consists of geodesics that form a bigon about the cusp; we express all such geodesics in terms of Markoff triples. We also show that any geodesic joining two elliptic fixed points has locally isolated height. 
Thomas A. Schmidt (Oregon State University), Mark Sheingorn (CUNY -   Baruch College)