On homogeneous 3-dimensional spacetimes: focus on plane waves

We revisit the classification of Lorentz homogeneous spaces of dimension $3$, and relax usual completeness assumptions. In particular, non-unimodular elliptic plane waves, and only them, are neither locally symmetric nor locally isometric to a left-invariant Lorentz metric on a $3$-dimensional Lie group. We characterize homogeneous plane waves in dimension $3$, and prove they are non-extendable, and geodesically complete only if they are symmetric. Finally, only one non-flat plane wave has a compact model.