ON THE INFINITE VOLUME HECKE SURFACES

An infinite volume Hecke surface, G(lambda)\H is the Riemann surface a ssociated with the Hecke triangle group of translation length lambda > 2. This paper: (i) gives an algorithm producing the length spectrum f or each of these surfaces employing an unramified double cover (by way of illustration, we tabulate the shortest 25 geodesics for the case l ambda = 4. We know of no other infinite volume surface for which this data exists). (ii) Establishes the existence of a Hall ray for open ge odesics on G(lambda)\H Our proof requires that lambda > root 8. The ex istence of a Hall ray means (after Haas) that the set consisting of th e highest penetration of each geodesic into the fundamental horocycle is dense in some real half-line. It is necessary to use open geodesics -there is no Hall ray otherwise. This is in marked contrast to the fin ite volume case, as we prove.