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Infra-Abelian groups and free actions of finite groups on the $$n$$-torus. (English) Zbl 1004.20027
We denote by $$\mathbb{Z}_n$$ the group of integers $$\text{mod }n$$, by $$\mathbb{Z}^h_n$$ the product of $$h$$ copies of $$\mathbb{Z}_n$$ and by $$[x]$$ the greatest integer less than or equal to $$x$$. The authors consider the problem of free actions of $$\mathbb{Z}^h_p$$, $$p$$ prime, on the $$n$$-torus $$T^n$$, which is related to the problem of torsion-free extensions: $0\to\mathbb{Z}^n\to\Gamma\to(\mathbb{Z}_p)^h\to 0.$ In particular they prove: 1. Let $$\pi$$ be a Bieberbach group of dimension $$n$$ with holonomy group $$\Phi$$. Then there is a faithful representation of $$\Phi$$ into $$\text{GL}(n-b_1,\mathbb{Z})$$ where $$b_1$$ denotes the first Betti number of the group $$\pi$$. In particular if $$\Phi=\mathbb{Z}^r_p$$ then $$r\leq[n-b_1/p-1]$$.
2. Suppose that $$\mathbb{Z}^h_p$$ acts freely on $$T^n$$ and faithfully on $$\pi_1(T^n)=\mathbb{Z}^n$$. Then in case $$b_1=0$$, $$1<h\leq[(n/p-1)+(1-p)]$$ and in case $$1\leq b_1<n$$, $$h\leq[n-b_1/p-1]$$ where $$b_1$$ denotes the first Betti number of $$T^n/\mathbb{Z}^h_p$$.
3. If $$\mathbb{Z}^h_p$$ acts freely on $$T^4$$ then $$h\leq 4$$.
4. If $$\mathbb{Z}^h_p$$ acts freely on $$T^5$$ and the center of $$\pi_1(T^5/\mathbb{Z}^h_p)$$ is not trivial then $$h\leq 5$$.
The reviewer’s opinion is that the most of results of the paper under review are related to the works: [H. Hiller and C. H. Sah, Q. J. Math., Oxf. II. Ser. 37, 177-187 (1986; Zbl 0598.57014); H. Zheng, Commun. Algebra 22, No. 4, 1403-1417 (1994; Zbl 0813.20041)], which is mentioned by the authors. To the above list we would like to add the article [E. Yalcin, Trans. Am. Math. Soc. 352, No. 6, 2689-2700 (2000; Zbl 0972.57023)].

##### MSC:
 20H15 Other geometric groups, including crystallographic groups 55N99 Homology and cohomology theories in algebraic topology 20C11 $$p$$-adic representations of finite groups
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