I would like to know if there exist examples on affine manifolds $M$ which are both complete and inhomogeneous.
Here completeness means that the universal cover $\widetilde{M}$ of $M$ is affinely equivalent to $\Bbb R^n$ (with the canonical structure) and homogeneity means that the group $\mathsf{Aff}(M)$ of affine automorphisms of $M$ acts transitively on $M$.
If $M$ is complete, it is affinely equivalent to a quotient $\Bbb R^n/\Gamma$ where $\Gamma$ is a subgroup of the affine group $$\mathsf{Aff}(n)=\{x\in\Bbb R^n\mapsto Ax+b\in\Bbb R^n;~A\in GL_n(\Bbb R),~b\in\Bbb R^n\}$$ acting by covering space action on $\Bbb R^n$. If I can find a subgroup $G$ of $\mathsf{Aff}(n)$ which acts transitively on $\Bbb R^n$ such that any $g\in G$ factors to a map $\tilde{g}:\Bbb R^n/\Gamma\to\Bbb R^n/\Gamma$, then $M$ is homogeneous. But I am pretty sure this is not always possible, that's why I am asking the question.
Thanks in advance for your help.