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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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