Hyperbolic structures for surfaces of infinite type

Abstract

Our main objective is to understand the geometry of hyperbolic structures on surfaces of infinite type. In particular, we investigate the properties of surfaces called flute spaces which are constructed from infinite sequences of "pairs of pants," each glued to the next along a common boundary geodesic. Necessary and sufficient conditions are supplied for a flute space to be constructed using only "tight pants," along with sufficient conditions on when the hyperbolic structure is complete. An infinite version of the Klein-Maskit combination theorem is derived. Finally, using the above constructions a number of applications to the deformation theory of infinite type hyperbolic surfaces are examined.