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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.

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If you want drawings, you need to see our recent work!

In HyperRogue 11.2 you can explore all the eight Thurston geometries.

See the blogpost for a simple description and some videos, and the geometries page for a longer description with some references. The easiest way to play (in 11.2b) is: menu -> special modes -> racing -> racing in Thurston geometries.

Also see a related question on MathOverflow.

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The Shape of Space by Jeffrey Weeks.

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

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

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