An invariant of free central extensions (Q1100288)

Let \(R\rightarrowtail F\twoheadrightarrow G\) be a free presentation of the finitely generated group \(G\), let \(T/[F,R]\) denote the torsion subgroup of \(R/[F,R]\) and set \(A=R/T\). The author considers the central extension \(A\rightarrowtail\Gamma\twoheadrightarrow G\) arising from the given presentation of \(G\), and the localization \(S^{-1}{\mathbb{C}}\Gamma\) of \({\mathbb{C}}\Gamma\) in which the elements of \(S={\mathbb{C}}A-\{0\}\) are inverted. It is shown that the global dimension of \(S^{-1}{\mathbb{C}}\Gamma\) is independent of the presentation of \(G\) and thus an invariant of \(G\), called \(\xi(G)\) by the author. The paper contains a number of results on this invariant. It is first proved that one has \(\xi(G)\leq cd_{{\mathbb{C}}}(G)\). In certain cases, for examples if \(G\) is a direct product of non-abelian free groups, this becomes an equality. In general however one has \(\xi(G)<cd_{{\mathbb{C}}}(G)\), as follows from the following somewhat surprising result: If \(G\) is a free abelian group, then \(\xi(G)=1\). The complicated proof of this hard result occupies almost half of the paper.