The space of vectored hyperbolic surfaces is path-connected (Q6622914)

In this interesting paper, the author studies the space \(\mathcal{H}^2\) of connected oriented complete hyperbolic surfaces of finite or infinite topological type without boundary decorated with a base unit vector, equipped with the topology induced by the Gromov-Hausdorff convergence. He calls an element of this space a \textit{vectored hyperbolic surface}. Roughly speaking, two vectored hyperbolic surfaces are close if there is a correspondence between points in large compact balls centered at the basepoints that respects the base vectors and that is close to an isometry. The author notes that this topology coincides with the Chabauty topology on the space of discrete torsion-free subgroups of \(\mathrm{PSL}(2,\mathbb{R})\). He also notes that \(\mathcal{H}^2\) is the 2-dimensional analogue of the space \(\mathcal{H}^n\) of framed hyperbolic \(n\)-manifolds equipped with the geometric topology that appears in the works of Jørgensen and Thurston.