** "Geometry and Analysis on Groups" Research Seminar **

In the first part, I will recall some notions from pseudo-Riemannian geometry and some basic properties of uniform and finite-covolume subgroups of Lie groups. The second part will be about recent results on the structure of the isometry groups of such spaces \(M\). Here, a strong invariance property for the indefinite metric on \(M\) can be derived from the finite invariant volume, and this allows us to significantly reduce the possible candidates for isometry groups. This leads to certain classification results for low metric index and for special geometries, such as manifolds \(M\) with pseudo-Einstein metrics or \(G_{2(2)}\)-structures.