Torsion-free groups with indecomposable holonomy group. I (Q2776802)

A classical crystallographic group \(\Gamma\) is a discrete cocompact subgroup of \(E(n)\), the isometry group of \(\mathbb{R}^n\). From the Bieberbach theorems there is a short exact sequence NEWLINE\[NEWLINE0\to\mathbb{Z}^n\to\Gamma\to G\to 0,NEWLINE\]NEWLINE where \(G\) is a finite group. It defines, by conjugation, a faithful holonomy representation \(\phi\colon G\to\text{GL}(n,\mathbb{Z})\). \(G\) is usually called a holonomy group of \(\Gamma\). From the homology point of view, \(\Gamma\) is also defined by a cohomology class \(\alpha\in H^2(G,\mathbb{Z}^n)\). Let \(\mathbb{Z}\), \(\mathbb{Z}_{(p)}\) and \(\mathbb{Z}_p\) denote, respectively, the ring of rational integers, the localization of \(\mathbb{Z}\) at the prime \(p\) and the ring of \(p\)-adic integers. For short, we define a generalized crystallographic group as an element in \(H^2(G,K^n)\), where \(K=\mathbb{Z}_{(p)},\mathbb{Z}_p\) and \(G\) acts faithfully on \(K^n\). The main results of the paper under review are related to classical and generalized crystallographic groups with cyclic holonomy group \(C_{p^s}\) of order \(p^s\) and holonomy group \(C_2\times C_2\). In particular it is proved that for \(s\geq 3\) the set of \(R\)-dimensions of the indecomposable \(RC_{p^s}\)-modules \(M\) for which there exist a torsion-free crystallographic group \(0\to M\to\Gamma\to C_{p^s}\to 0\) is unbounded. Here \(R=\mathbb{Z},\mathbb{Z}_{(p)},\mathbb{Z}_p\).NEWLINENEWLINENEWLINEMoreover for \(s=2\) is given a classification, up to isomorphism and in terms of indecomposable modules and cocycles, of torsion-free generalized crystallographic groups with indecomposable holonomy representation. For the classical case it means that there exist at least \(2p-3\) Bieberbach groups with cyclic indecomposable holonomy group of order \(p^2\).NEWLINENEWLINENEWLINEThe last main result gives a table of torsion-free crystallographic groups with holonomy group \(C_2\times C_2\) and indecomposable holonomy representation. This classification also considers the generalized case for \(K=\mathbb{Z}_{(2)},\mathbb{Z}_2\).