Exterior powers of modules for group rings of polycyclic groups.

*(English)*Zbl 0911.20006The authors prove that every finitely presented abelian-by-polycyclic group has a metanilpotent normal subgroup of finite index (Theorem A), so answering in the negative the question (1973) by G. Baumslag, whether every finitely generated (f.g.) abelian-by-polycyclic group can be embedded into a finitely presented (f.p.) group in the same class. This question of G. Baumslag [Proc. 2nd internat. Conf. Theory of Groups, Canberra 1973, Lect. Notes Math. 372, 65-74 (1974; Zbl 0304.20015)] has been the primary motivation for this paper.

Theorem A is deduced from the following fact (Theorem B): Let \(H\) be a polycyclic group and \(M\) a f.g. \(\mathbb{Z} H\)-module. If the exterior power \(\bigwedge^r M\) (for some \(r\geq 2\)) is f.g., regarded as a \(\mathbb{Z} H\)-module with \(H\) acting diagonally, then the factor-group of \(H\) by the unique largest normal subgroup \(\eta_H(M)\) of \(H\) which acts nilpotently on \(M\), is virtually nilpotent. To prove Theorem B the authors suppose it to be false and take \(H\) to be a polycyclic group of the least possible Hirsch number for which the assertion of Theorem B fails. This last holds then also for all subgroups of finite index in \(H\), so allowing to further suppose that \(H\) is an orbitally sound, torsion-free polycyclic group, whose centre coincides with its FC-centre. It follows that \(C_H(M)=1\) and \(M\) is a prime \(\mathbb{Z} A\)-module for each abelian normal subgroup \(A\) in \(H\). Using J. Roseblade’s control theorems [Proc. Lond. Math. Soc., III. Ser. 36, 385-447 (1978; Zbl 0391.16008)] for prime ideals in group algebras of polycyclic groups, this argument is further reduced to consideration of polycyclic groups \(H\) with \(A\) a self-centralizing plinth of \(H\) whose rank is greater than 1, with \(Q=H/A\) being free abelian and with \(\eta_H(M)=1\). Here, according to J. Roseblade [(*) J. Pure Appl. Algebra 3, 307-328 (1973; Zbl 0285.20008)], \(A\) is a plinth of \(H\) if \(A\otimes_{\mathbb{Z}}\mathbb{Q}\) when treated as a \(\mathbb{Q} H_0\)-module is irreducible for every subgroup \(H_0\) of finite index in \(H\). It follows that \(Q\) embeds into the unit group of some number field and the Dirichlet unit theorem gives \(r_0(Q)<r_0(A)\) for ranks. Then, the following result (Theorem C) is used: Let \(G\) be a split extension of a non-trivial f.g. free abelian group \(A\) by a f.g. free abelian group \(Q\) and let \(\mathbf k\) be a locally finite field. Further, let \(S\) be a \(Q\)-invariant subring of \({\mathbf k}A\) and suppose that there exists a module for \(SQ\) – this last being the subring generated by \(S\) and \(Q\) – which is f.g. for \({\mathbf k}Q\) and not torsion for \(S\). If \({\mathbf k}A\) is integral over \(S\), then \(r_0(Q)\geq r_0(A)\). Supposing that Theorem C holds, together with the supposition that \(r\geq 2\), \({\mathbf k}\) is a finite field, \(M\) is a f.g. \({\mathbf k}H\)-module which is torsion-free as a \({\mathbf k}A\)-module of rank \(\geq r\) and \(\bigwedge^r M\) is f.g. as a \({\mathbf k}H\)-module, the authors prove \(r_0(Q)\geq r_0(A)\). This contradicts \(r_0(Q)<r_0(A)\) above and so giving Theorem C \(\Rightarrow\) Theorem B.

At last, it is proved Theorem C using J. Roseblade’s theorem C* [in (*)] and \(\dots\) the theory of Hilbert-Serre dimension. For interesting details and wealth of ideas used in this truely remarkable paper on group rings the reader should consult the paper itself.

Theorem A is deduced from the following fact (Theorem B): Let \(H\) be a polycyclic group and \(M\) a f.g. \(\mathbb{Z} H\)-module. If the exterior power \(\bigwedge^r M\) (for some \(r\geq 2\)) is f.g., regarded as a \(\mathbb{Z} H\)-module with \(H\) acting diagonally, then the factor-group of \(H\) by the unique largest normal subgroup \(\eta_H(M)\) of \(H\) which acts nilpotently on \(M\), is virtually nilpotent. To prove Theorem B the authors suppose it to be false and take \(H\) to be a polycyclic group of the least possible Hirsch number for which the assertion of Theorem B fails. This last holds then also for all subgroups of finite index in \(H\), so allowing to further suppose that \(H\) is an orbitally sound, torsion-free polycyclic group, whose centre coincides with its FC-centre. It follows that \(C_H(M)=1\) and \(M\) is a prime \(\mathbb{Z} A\)-module for each abelian normal subgroup \(A\) in \(H\). Using J. Roseblade’s control theorems [Proc. Lond. Math. Soc., III. Ser. 36, 385-447 (1978; Zbl 0391.16008)] for prime ideals in group algebras of polycyclic groups, this argument is further reduced to consideration of polycyclic groups \(H\) with \(A\) a self-centralizing plinth of \(H\) whose rank is greater than 1, with \(Q=H/A\) being free abelian and with \(\eta_H(M)=1\). Here, according to J. Roseblade [(*) J. Pure Appl. Algebra 3, 307-328 (1973; Zbl 0285.20008)], \(A\) is a plinth of \(H\) if \(A\otimes_{\mathbb{Z}}\mathbb{Q}\) when treated as a \(\mathbb{Q} H_0\)-module is irreducible for every subgroup \(H_0\) of finite index in \(H\). It follows that \(Q\) embeds into the unit group of some number field and the Dirichlet unit theorem gives \(r_0(Q)<r_0(A)\) for ranks. Then, the following result (Theorem C) is used: Let \(G\) be a split extension of a non-trivial f.g. free abelian group \(A\) by a f.g. free abelian group \(Q\) and let \(\mathbf k\) be a locally finite field. Further, let \(S\) be a \(Q\)-invariant subring of \({\mathbf k}A\) and suppose that there exists a module for \(SQ\) – this last being the subring generated by \(S\) and \(Q\) – which is f.g. for \({\mathbf k}Q\) and not torsion for \(S\). If \({\mathbf k}A\) is integral over \(S\), then \(r_0(Q)\geq r_0(A)\). Supposing that Theorem C holds, together with the supposition that \(r\geq 2\), \({\mathbf k}\) is a finite field, \(M\) is a f.g. \({\mathbf k}H\)-module which is torsion-free as a \({\mathbf k}A\)-module of rank \(\geq r\) and \(\bigwedge^r M\) is f.g. as a \({\mathbf k}H\)-module, the authors prove \(r_0(Q)\geq r_0(A)\). This contradicts \(r_0(Q)<r_0(A)\) above and so giving Theorem C \(\Rightarrow\) Theorem B.

At last, it is proved Theorem C using J. Roseblade’s theorem C* [in (*)] and \(\dots\) the theory of Hilbert-Serre dimension. For interesting details and wealth of ideas used in this truely remarkable paper on group rings the reader should consult the paper itself.

Reviewer: U.Kaljulaid (Tartu)

##### MSC:

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

16S34 | Group rings |

20C12 | Integral representations of infinite groups |

20F19 | Generalizations of solvable and nilpotent groups |

20F05 | Generators, relations, and presentations of groups |

20E07 | Subgroup theorems; subgroup growth |

20F16 | Solvable groups, supersolvable groups |