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Exterior powers of modules for group rings of polycyclic groups. (English) Zbl 0911.20006
The 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.

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
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