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Homological properties of abstract and profinite modules and groups. (English) Zbl 1170.20031

The author [in J. Pure Appl. Algebra 204, No. 3, 536-554 (2006; Zbl 1089.20031), Comment. Math. Helv. 81, No. 4, 931-943 (2006; Zbl 1166.20042)] proved a conjecture due to E. Rapaport Strasser about knot-like groups, i.e., if a knot-like group \(G\) has a finitely generated commutator subgroup \(G'\), then \(G'\) should be free. A knot-like group is a finitely presented of deficiency one group with infinite cyclic Abelianization. A pro-\(p\) version of the Rapaport Strasser Conjecture was suggested and resolved [in op. cit., Zbl 1089.20031], and a profinite version of the same conjecture was proved by F. Grunewald, A. Jaikin-Zapirain, A. G. S. Pinto, P. A. Zalesski [in Normal subgroups of profinite groups on non-negative deficiency (preprint)].
In this paper the author generalizes the main results of Grunewald et al. and op. cit. [Zbl 1166.20042] about cohomological dimensions and the homological type \(\text{FP}_m\) of a normal subgroup of a group \(G\) with some finiteness properties (in both discrete and profinite cases) and treats the module versions of the same results.

MSC:

20J05 Homological methods in group theory
57M07 Topological methods in group theory
20F05 Generators, relations, and presentations of groups
20E18 Limits, profinite groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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