The space of vectored hyperbolic surfaces is path-connected

In the space H 2 H^2 of hyperbolic surfaces decorated with a base unit vector, the topology induced by the Gromov–Hausdorff convergence coincides with the Chabauty topology on the space of discrete torsion-free subgroups of P S L 2 ( R ) PSL_2(R) . Using paths constructed from changing the Fenchel–Nielsen coordinates and shrinking simple closed curves to cusps, we demonstrate path-connectivity of H 2 H^2 and some of its subspaces.