Exotic Geometries

Question

I only recently realised there are more geometries than euclidean, spherical, hyperbolic and mix of them. But the wikipedia page on The eight Thurston geometries in 3 dimensions is cryptic to me. I'm looking for nice references that describe metric spaces with associated geodesics and properties for the geometry of the universal cover of $SL(2, R)$, the Nil geometry and the Solv geometry. With drawings !

For example, the book of J. Ratcliffe 'Foundations of hyperbolic manifolds' is a nice start for hyperbolic geometry, I'd like something similar for those exotic geometries.

Answer 1

The Shape of Space by Jeffrey Weeks.

Answer 2

Three dimensional geometry and topology by William P Thurston is a nice book with drawings

Answer 3

Also I'd point out this recent article where the Thurston geometries are rendered from within!