While the geometric theory of finite type surfaces is well developed, the study of hyperbolic geometric structures on infinite type surfaces (that is, infinitely generated fundamental group) is still in its infancy. On the other hand, the (complex analytic) theory of infinite type Riemann surfaces, for example plane domains, is much more advanced.

In this talk we will first describe some of the known results on the geometry and topology of such surfaces and then discuss new results involving the relationship between Fenchel-Nielsen coordinates and a version of the classical type problem (whether or not the surface carries a Green's function). In particular, we study so called tight flute surfaces - possibly incomplete hyperbolic surfaces constructed by linearly gluing infinitely many tight pairs of pants along their cuffs - and the relationship between their type and geometric structure. This is joint work with Dragomir Saric. Speaker(s): Ara Basmajian (City University of New York)