When does an affine manifold inherit a (quotient) group action?

Question

By an affine manifold I mean a real $n$-dimensional manifold $M$ with charts whose transition functions are in the affine group $Aff(\Bbb R^n)$. There are several other equivalent definitions https://en.wikipedia.org/wiki/Affine_manifold. An affine manifold is called complete if its universal covering is homeomorphic to $\Bbb R^n$. One way to construct complete affine manifolds is just to take $M= \Bbb R^n/\Gamma$, where $\Gamma \subset Aff(\Bbb R^n)$ is a discrete subgroup whose action (inherited from the $Aff(\Bbb R^n)$ action) on $\Bbb R^n$ is free and properly discontinuous. It follows that $M$ has fundamental group $\Gamma$. What can we say about possible group actions on an $M$ constructed in this way? Is it true that $M$ always inherits an action of $Aff(\Bbb R^n)/\Gamma$ provided that $\Gamma \subset Aff(\Bbb R^n)$ is normal, and if so is this the only way to realise a non-trivial group action? Examples would be appreciated.

Answer 1

Just a remark, it is proved in Wolf's book "Spaces of constant curvature" that every compact complete affine manifold arises as a quotient $M=\mathbb{R}^n/\Gamma$, where $\Gamma\le Aff(\mathbb{R}^n)$ acts properly discontinuously and cocompactly on $\mathbb{R}^n$.
Then there is a famous conjecture about such group actions, called Auslander conjecture:

Conjecture (Auslander 1964): A group $\Gamma\le Aff(\mathbb{R}^n)$ acting properly discontinuously and cocompactly on $\mathbb{R}^n$ is virtually polycyclic.

There is a lot of literature about affine actions on Lie groups (and in particular on affine manifolds).