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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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