Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group? 


Edit: My goal is to understand a bit better what the intuition behind compactness / non-compactness of the isometry of a closed Semi-Riemannian manifold is. For example are there topological restrictions on the underlying space? An example of the kind of statements I'm looking for is the following theorem (for Lorentzian manifolds):


**Theorem:** Let $(M,g)$ be a closed, simply connected Lorentz manifold and assume $M$ and $g$ are analytic. Then the isometry group of $M$ is compact.



Closed Semi-Riemannian manifolds with non-compact isometry group
Of course there is D’Ambra's 1988 paper "Isometry groups of Lorentz manifolds", from which the theorem you state is taken.
A later paper, taking a more general perspective, is [this one](http://www.ihes.fr/~/gromov/PDF/1[74].pdf) by D’Ambra and Gromov from 1991.


Zimmer's school had its impact then, with Kowalski's thesis from 1994 and further papers by Zimmer, Adams, Stuck, Witte-Morris and Iozzi.


Later contribution was made by Zeghib's school and in particular Charles Frances.


There are many other contributions. Take a look at the list of participants of [this conference](http://cs.slu.edu/~scannell/lorentz/).
Many of the later contributions are in the language of Cartan geometries or rigid geometric structures etc. so you might miss them by a careless literature briefing.


My advice it to look at some of Frances' late papers, such as [this](https://arxiv.org/pdf/1804.08695.pdf), and the references there.
Maybe even better: make a contact with Frances who is actively working on the subject.




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Let me add just a little bit of actual mathematics.


Let $G$ be a semisimple Lie group. Then its Killing form is an $\text{Ad}$-invariant non-degenerate symmetric form on its Lie algebra, which I identify here the tangent space at the identity.
It is definite iff $G$ is compact.
Applying left translations we obtain a pseudo-Riemannian structure on $G$ which is both left and right translation invariant. 
Picking a cocompact lattice $\Gamma<G$ we form a quotient manifold $G/\Gamma$ and endow it with the quotient pseudo-Riemannian structure (note that $G\to G/\Gamma$ is a covering map). The left action of G on this space will be by isometries.
The particular example of $G=\text{SL}\_2(\mathbb{R})$ will give us a 3-dimensional compact Lorentzian manifold endowed with a non-trivial continuous action of $\text{SL}\_2(\mathbb{R})$ by isometries.
In a sense, this is as large as a group of isometries of compact Lorentzian manifold can get: non-compact simple Lie groups other then (locally) $\text{SL}\_2(\mathbb{R})$ have no non-trivial continuous action of on compact Lorentzian manifolds.
Similar theorems (with larger sets of exceptions) could be proved for pseudo-Riemannain manifolds with higher signatures.


