Let a \(n\)-dimensional compact flat manifold. It is well described by Bieberbach, who characterized this manifold as being isometrically covered by a flat torus, with its fundamental group \(\Gamma \) torsion-free and having a maximal abelian normal subgroup \(M\) of finite index [L. S. Charlap, Bieberbach groups and flat manifolds. New York etc.: Springer Verlag (1986; Zbl 0608.53001)]. The group \(\Gamma\) is called an \(n\)-dimensional Bieberbach group. The quotient \(G\,=\,\Gamma /M\) called the holonomy group, is a finite group acting faithfully on \(M\).
The investigation of the extent to which a Bieberbach group may be distinguished from another by its set of finite quotient groups, in the general case, is a difficult problem, since there is no complete classification of Bieberbach groups in all dimensions.
In [J. Group Theory 24, No. 6, 1135–1148 (2021; Zbl 1514.20128)] the author gives a complete answer to this problem in the case that the holonomy group is cyclic of prime order. In this paper it is considered the Bieberbach groups \(\Gamma\) with cyclic holonomy group \(G\,=\,C_{p_{1}}\times C_{p_{2}} \times \,\cdots \,\times C_{p_{k}}\), where \(C_{p_{i}}\) is a cyclic group of prime order \(p_{i}\). Thus, this paper can be considered as a natural continuation of the previous work.
Let \(M\) be a faithful \(\mathbb{Z}G\)-lattice for a finite group \(G\). The crystal class \((G,\,M)\) is defined to be the the set of all free-torsion extensions \(\Gamma\) of \(G\) by \(M\). Two crystal classes \((G,\,M)\) and \((G^{\prime},\, M^{\prime})\) are arithmetically equivalent if \(G\) and \(G^{\prime}\) are conjugate subgroups of \(GL(n,\,\mathbb{Z})\). The resulting equivalence classes are the arithmetic crystal classes. Let \(\mathcal{C}(M)\) denote the set of isomorphism classes of the \(\mathbb{Z}G\)-lattice \(N\), which correspond to the arithmetic crystal classes of \((G,\,N)\), such that the \(\hat{\mathbb{Z}}G\)-modules \(\hat{N}\) and \(\hat{M}\) are isomorphic.
The cardinality \(\mid \mathcal{C}(M)\mid\) is calculated in Theorem 1.2 of the paper.
The genus \(g(\Gamma )\) is defined as the set of isomorphism classes of finitely generated residually finite groups with the profinite completion isomorphic to the profinite completion \(\hat{\Gamma}\) of \(\Gamma\).
Let now \(\Gamma\) be a Bieberbach group with cyclic holonomy group \(G\,=\,C_{p_{1}}\times C_{p_{2}} \times \,\cdots \,\times C_{p_{k}}\), where \(C_{p_{i}}\) is a cyclic group of prime order \(p_{i}\).
The main aim of this paper (Theorem 1.3) is to find a formula for the genus of this Bieberbach group \(\Gamma\).
To calculate the cardinality \(\mid g(\Gamma )\mid \) it is proved that to find all possible Bieberbach groups (up to isomorphism) of an arithmetic crystal class, it is sufficient to find all possible Bieberbach groups of only one representative of the class (Remark 2.19). This means that it is sufficient to consider a set \(T\) of representatives for the isomorphism classes of \(\mathbb{Z}G\)-lattices \(N\) in \(\mathcal{C}(M)\). Theorem 1.2 gives a formula for the cardinality of \(T\).
As an immediate corollary, the main result in [J. Group Theory 24, No. 6, 1135–1148 (2021; Zbl 1514.20128)] is obtained.
Let \(\Gamma \) be an \(n\)-dimensional Bieberbach group with maximal abelian normal subgroup \(M\) and cyclic holonomy group \(G\) of square-free order. If \(\mid G\mid\) is a prime number, then \(\mid g(\Gamma )\mid\,=\,\mid \mathcal{C}(M)\mid\). (Corollary 1.6).
In particular, the author proves the following:
Let \(\Gamma \) be an \(n\)-dimensional Bieberbach group with the cyclic holonomy group of order \(\delta , = \,6, 10\) or 14. Then \(\mid g(\Gamma )\mid\,=\,1\). (Theorem 1.7).