# Learning roadmap for Lorentzian geometry

I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF.

I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (such as hyperbolic surfaces, Teichmuller theory). I have a background in 2-dimensional hyperbolic geometry (S. Katok, Beardon), and 3-dimensional hyperbolic geometry (basic stage), (basic) algebraic topology (Munkres), Riemann surfaces, smooth manifolds, (basic) Riemannian geometry. I want to study more on hyperbolic geometry towards the following direction: the relation between Anti-de Sitter geometry and Teichmuller theory (see here). I want to learn Anti-de sitter geometry.

I found that Anti-de sitter geometry intersects (may be a subtopic) with Lorentzian $$(n; 1)$$ geometry. Moreover, one can examine the constant curvature spaces: flat, de Sitter (positive curvature) and anti-de Sitter (negative curvature) spaces in context of Lorentzian geometry. So, I think anti-de Sitter might be similar type of hyperbolic geometry. Also, I have heard that (flat) Lorentz $$(3; 1)$$ space is the natural home for Einstein's Special Theory of Relativity, with three "space" dimensions and one "time" dimension. But, I am not a student from physics background. I want to study the hyperbolic geometric aspects of the subject. I am eager to learn Lorentzian geometry and anti-de Sitter geometry thoroughly and rigorously. Also, I want to learn its relations with Teichmuller theory (see here, Prop. 22 for the Earthquake Theorem).

I did not find any books related to the above said topics e.g. Lorentzian geometry and anti-de Sitter geometry. I scouted the internet and found some papers on the topics such as "Crooked surfaces and anti-de Sitter geometry" (see here), "The geometry of crooked planes" (see here), "Fundamental polyhedra for Margulis space-times" (see here), "The Margulis Invariant of Isometric Actions on Minkowski $$(2+1)$$-Space" (see here), etc. But I do not know about the proper way to study the subject. Also, I found that those papers discussing Lorentzian geometry (anti-de Sitter geometry, in particular ) also discuss about Einstein universe, crooked planes, affine transformations, proper affine actions, Margulis space-times, Margulis Invariant, crystallographic groups, etc. I have no idea about the previous said keywords, e.g. crooked planes, Margulis space-times. I want to know why one should discuss crooked planes, Margulis space times to learn Lorentzian geometry (in particular, anti-de Sitter geometry).

I want to learn Lorentzian geometry and anti-de Sitter geometry thoroughly and rigorously. Then I want to focus on its relation with Teichmuller theory (may be later).