Connectivity of the space of pointed hyperbolic surfaces

Abstract

We consider the space $\rootedH2$ of all complete hyperbolic surfaces without boundary with a basepoint equipped with the pointed Gromov-Hausdorff topology. Continuous paths within $\rootedH2$ arising from certain deformations on a hyperbolic surface and concrete geometric constructions are studied. These include changing some Fenchel-Nielsen parameters of a subsurface, pinching a simple closed geodesic to a cusp, and inserting an infinite strip along a proper bi-infinite geodesic. We then use these paths to show that $\rootedH2$ is path-connected and that it is locally weakly connected at points whose underlying surfaces are either the hyperbolic plane or hyperbolic surfaces of the first kind.