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Free actions of Abelian \(p\)-groups on the \(n\)-torus. (English) Zbl 1075.20023
Let \(H\) be a finite group acting freely on the torus \(T^n\). For a given group \(H\), the question of finding the minimal value of \(n\) is considered in the paper under review. The authors try to estimate such number \(n\), for \(H\) a \(p\)-group, under the hypothesis that the induced representation is faithful and the first Betti number \(b\) is different from zero. That is equivalent to give the minimal dimension of a Bieberbach group with given, not equal to zero, rank of its Abelianization and given holonomy. For the case of the first Betti number equal to zero this problem was considered by H. Zheng [Commun. Algebra 22, No. 4, 1403-1417 (1994; Zbl 0813.20041)].
The main result is following. Let \(\delta(H,b)\) denote the minimal dimension of a Bieberbach group with holonomy group \(H\) and the first Betti number equal to \(b\).
Theorem: Suppose \(H=\mathbb{Z}^{h_1}_{p^{k_1}}\times\mathbb{Z}^{h_2}_{p^{k_2}}\times \cdots\times\mathbb{Z}^{h_r}_{p^{k_r}}\) for \(r\geq 1\), \(k_1\geq k_2\geq\cdots\geq k_r\geq 1= k_{r+1}\). Then \(H\) acts freely and faithfully on \(T^n\) for \[ n=\sum^r_{i=1}[(h_i-1)(p^{k_i}-1)+\varphi(p^{k_i})+(p^{k_{i+1}-1}-1)] +b, \] with the first Betti number \(b\geq 1\) and \(n=\sum^r_{i=1} h_i\varphi(p^{k_i})+b\), with first Betti number \(b\geq h_1+h_2+\cdots+h_r\). Moreover, \[ \sum^r_{i=1}[(h_i-1)(p^{k_i}-1)+\varphi(p^{k_i})+ (p^{k_{i+1}-1}-1)]+b\geq\delta(H,b) \] and \(\delta(H,b)\geq\sum^r_{i=1} h_i\varphi(p^{k_i})+b\), where \(\varphi\) denotes the Euler function. Furthermore, if \(b=h_1+h_2+\cdots+h_r\), then we have \(\delta(H,b)=\sum^r_{i=1}h_i\varphi(p^{k_i})+b\).
The authors generalize the notion of a faithful action to the space \(X=S^m\times\cdots\times S^m\), where \(m\geq 1\). They understand it as a faithful action on \(H_m(X,\mathbb{Z})\). If a finite group \(H\) acts on \(X\) then the first Betti number of the action is the rank of the space \((H_m(X,\mathbb{Z}))^H\). In that case the following theorem is proved.
Theorem: Suppose \(H=(\mathbb{Z}_2)^h\). Then \(H\) acts freely and faithfully on \((S^m)^n\) for the values \(n=h+b\) if either \(m\) is even and the first Betti number \(b\geq 0\) or if \(m\) is odd and the first Betti number \(b\geq 1\). For \(m\) odd and the first Betti number \(b=0\) then \(H\) acts freely and faithfully on \((S^m)^{h+1}\) for \(h>1\) and \(\mathbb{Z}_2\) does not act freely and faithfully on \((S^m)^2\) with \(b=0\). Moreover, \(\delta(H,b)=h+b\) for either \(m\) even, or \(m\) odd and \(b\geq 1\). For \(b=0\) and \(m\) odd we have \(\delta(H,0)=h+1\) if \(h>1\).
There are given many examples of free and faithful actions on products of spheres.
MSC:
20H15 Other geometric groups, including crystallographic groups
55M20 Fixed points and coincidences in algebraic topology
20K01 Finite abelian groups
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57S17 Finite transformation groups
57S25 Groups acting on specific manifolds
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