On Lorentz spacetimes of constant curvature

We describe in parallel the Lorentzian homogeneous spaces $G=\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ and $\mathfrak{g}=\mathfrak{p}\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$, and review some recent results relating the geometry of their quotients by discrete groups.