Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \mathbb{R}^k < {Aff}(\mathbb{R}^k)$, where $k < n$, how can we describe $M$ and $\pi_1(M)$? In particular, can we deduce that $M$ is homeomorphic to a torus, up to a finite cover?

There is a possible solution. Since $M$ is closed manifold, $\pi_1(M)$ is a finitely generated subgroup of $\Gamma(k)=O(k) \ltimes \mathbb{R}^k $. If $\pi_1(M)$ is discrete then we can apply Bieberbach theorem and deduce that $\pi_1(M)$ is virtually Abelian.

The question arose when I was contemplating a smooth codimension $k$ foliation with a transversal Euclidean structure, whose leaves are diffeomorphic to $\mathbb{R}^{n-k}$. The developing map yields the homomorphism $h: \pi_1(M) \rightarrow Iso(\mathbb{R}^k)$, and I am interested in knowing whether this implies that the manifold can only be a finite quotient of a torus.

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