I am looking for an introduction to hyperbolic surfaces as a quotient of the upper half plane by lattices.

Question

I keep coming across results of the form: If we take the quotient of the upper half plane by a Fuchsian group with this property, we get a surface with that property (cusps, funnels, in/finite area, ...).

Could you recommend a reference for those kinds of results?

Answer 1

Here are a few references to get you started:

• Thurston's notes and the book form, Geometry and Topology of 3-Manifolds

• Introduction to Teichmuller Theory, by Hubbard

• Lectures on Hyperbolic Geometry, by Benedetti and Petronio