The Space of Vectored Hyperbolic Surfaces is Path-Connected

Abstract

In the space ${\mathcal{H}}^2$ 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 ${\mathrm{PSL}}_2(\mathbb{R})$. Using paths constructed from changing the Fenchel-Nielsen coordinates and shrinking simple closed curves to cusps, we demonstrate path-connectivity of ${\mathcal{H}}^2$ and some of its subspaces.

11 pages, 3 figure; substantially corrected and edited from the previous version; accepted, to appear in Conformal Geometry and Dynamics