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Homological realization of prescribed Abelian groups via \(K\)-theory. (English) Zbl 1124.20036
A strongly torsion generated group \(G\) is one with the property that, for each \(n\geq 2\) there is an element \(g_n\) of order \(n\) such that every element of \(G\) is a product of conjugates of \(g_n\). Here the authors show: Theorem A. Let \(A\), \(B\) and \(C\) be any three Abelian groups. Then, there exists a group \(S\) with the following properties: (i) \(S\) is strongly torsion generated; (ii) the centre of \(S\) is isomorphic to \(A\); (iii) \(S\) is perfect; (iv) \(H_2(S,\mathbb{Z})\simeq B\); (v) \(H_3(S,\mathbb{Z})\simeq C\).
Various properties of the class of strongly torsion generated groups are discussed by A. J. Berrick and C. F. Miller, III. [in Math. Proc. Camb. Philos. Soc. 111, No. 2, 219-229 (1992; Zbl 0762.20017)]. It was introduced by A. J. Berrick [J. Algebra 139, No. 1, 190-194 (1991; Zbl 0745.20031)] because its most notable examples arise in connection with algebraic \(K\)-theory. They include the infinite alternating group \(A_\infty\) and the infinite special linear groups \(\text{SL}(\mathbb{Z})\) and \(\text{SL}(K)\) for any field \(K\). The proof of Theorem A uses algebraic and topological \(K\)-theory together with complex \(C^*\)-algebras.
The authors provide a nice review which indicates that past displays of Abelian groups as homology groups have formed separate results from realizations of Abelian groups as centres. In Theorem A they are able to combine these two strands in a single realization theorem.

MSC:
20J05 Homological methods in group theory
20K40 Homological and categorical methods for abelian groups
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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