I'm reading through the paper On the Geometry of Hurwitz Surfaces for an undergraduate project. Apologies for my basic questions; I've not met these ideas before.
In the abstract, the source says:
By uniformization, the surface admits a hyperbolic structure wherein the automorphisms act by isometry. Such isometries descend from the $(2,3,7)$ triangle group $T$ acting on the universal cover $\mathbb{H}^2$.
I understand by uniformisation, the only possible universal covers for a Riemann surface are the open unit disk, the complex plane, or the Riemann sphere. But how do we know for Hurwitz surfaces, the universal cover is the unit disk and so the surface is hyperbolic? (I assume this is the hyperbolic structure they are referring to?).
Furthermore, how do we know the isometries descend from the $(2,3,7)$ triangle group?
Please share any relevant introductory resources, as these will be of great help too.