Torsion-free metabelian groups with commutator quotient \(C_{p^n}\times C_{p^m}\)
An \(n\)-dimensional Bieberbach group \(G\) is a discrete, torsion-free, cocompact group of isometries of the \(n\)-dimensional Euclidean space. The corresponding orbit space \(M\), with \(G\) as fundamental group, is a (compact) flat Riemannian manifold. Conversely, each such manifold arises in this way. Algebraically, \(G\) is a torsion-free group containing a finitely generated maximal Abelian subgroup \(V\) (the translation subgroup) of finite index. The quotient \(G/V\) is called the point-group (or holonomy group) of \(G\). It is well known that \(G\) has a finite commutator quotient if and only if the first Betti number \(b_1(M)=0\) or equivalently, if and only if \(G\) is centerless [\textit{H. Hiller, C.-H. Sah}, Q. J. Math., Oxf. II. Ser. 37, 177-187 (1986; Zbl 0598.57014)]. In a previous work [\textit{C. F. Cid}, J. Group Theory 3, No. 2, 113-125 (2000; Zbl 0952.20040)], a specific family of centerless (metabelian) Bieberbach groups with point-group \(C_{p^n}\times C_{p^m}\), denoted by \(K(p^n,p^m)\), was introduced and studied. In this paper, considering the quotients of \(K(p,p^m)\), the author proves the existence of a finitely generated torsion-free metabelian group \(G\) with commutator quotient isomorphic to \(C_{p^n}\times C_{p^m}\) if and only if \(n,m\geq 2\) (Theorem A). The calculation of the torsion-free quotients of \(K(p,p^2)\) results in a complete list of torsion-free quotients of \(K(2,8)\) and \(K(4,4)\). From this it can be deduced that \(K(2,2)\), which is the fundamental group of the Hantzsche-Wendt manifold, is (up to isomorphism) the only finitely generated, torsion-free metabelian group with commutator quotient isomorphic with \(C_4\times C_4\).
finitely generated maximal Abelian subgroups
scientific article; zbMATH DE number 1763565