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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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    $\begingroup$ Two small comments: first, I think your semi-direct produce is the wrong way round. (Note that $\mathrm{Isom}(\mathbb{R}^k)\cong O(k)\ltimes \mathbb{R}^k$.) And second, do you really mean that $k<n$ rather than $k\leq n$? For instance, the $(3,3,3)$ triangle group is not a subgroup of the isometry group of $\mathbb{R}$ (since it has 3-torsion). $\endgroup$
    – HJRW
    Oct 12, 2023 at 12:57
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    $\begingroup$ Yes, some toral bundles over surfaces of arbitrary high dimension have fundamental groups embeddable in $SE(3)$. But why do you care? $\endgroup$ Oct 12, 2023 at 15:33
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    $\begingroup$ @HJRW Right. Then it is enough to observe that $\mathbf{Z}^n$ embeds into $\mathrm{SO}(2)$ for every $n$. $\endgroup$
    – YCor
    Oct 12, 2023 at 15:49
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    $\begingroup$ @HJRW Yes. Actually $\mathrm{SO}(3)$ itself contains large groups. Namely, consider an irreducible rational degree $n$ polynomial with $n$ real roots, only one of which, $r$ is positive, and $q$ the quadratic form $X^2+Y^2+rZ^2$. Then $\mathrm{SO}(q)(\mathbf{Z})$ is a dense subgroup of $\mathrm{SO}(3)$, and is also a cocompact lattice in $\mathrm{SO}(2,1)^{n-1}$ (hence virtual cohomological dimension $2n-2$, etc). $\endgroup$
    – YCor
    Oct 13, 2023 at 7:37
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    $\begingroup$ @Carl-FredrikNybergBrodda: This paper of Breuillard--Gelander--Souto--Storm provides a reference -- arxiv.org/abs/math/0602635 -- although I suspect it was known long before. I think their argument is roughly this one: for each $g\in\pi_1S$, the subset of $\mathrm{Hom}(\pi_1S,SO(3))$ that kills $g$ is nowhere dense, and the result follows by the Baire category theorem. Perhaps something like YCor's argument can give an explicit arithmetic construction. $\endgroup$
    – HJRW
    Oct 13, 2023 at 11:58

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