Homological and topological properties of locally indicable groups
This is an important paper: The authors show that three classes of groups, which have each been introduced in different contexts and for different reasons, actually coincide. This simplifies the existing theory and sheds new light on work of Adams on the Whitehead problem concerning asphericity in 2-complexes. The three classes of groups are the locally indicable groups, introduced by \textit{G. Higman} in connection with the zero divisor and unit problem in group rings [Proc. Lond. Math. Soc., II. Ser. 46, 231-248 (1940; Zbl 0025.24302)], the conservative groups, introduced by \textit{J. F. Adams} in connection with the Whitehead problem [J. Lond. Math. Soc. 30, 482-488 (1955; Zbl 0064.415)], and the Ḏ-groups introduced by \textit{R. Strebel} in connection with homological methods applied to the derived series of groups [Comment. Math. Helv. 49, 302-332 (1974; Zbl 0288.20066)]. Let \(R={\mathbb{Z}}\) or \({\mathbb{F}}_ p\). A group G is said to be locally R-indicable if every non-trivial finitely generated subgroup H admits an epimorphism \(H\twoheadrightarrow R\). G is conservative over R if \(H_ 2(K;R)=0\) for every 2-complex K acted on by G with \(H_ 2(K/G;R)=0.\) And G is a Ḏ(R)-group if \(R\otimes_{RG}\phi\) is non- injective whenever \(\phi:P\to Q\) is a non-injective homomorphism between projective RG-modules. (\(R\otimes_{RG}-\) detects injectivity). The main result, then, asserts that to be local R-indicable, to be conservative over R, and to be a Ḏ(R)-group are equivalent properties. There are also more subtle results comparing the three properties for different rings R. Applications. 1) For any group G let \(r(G)\triangleleft G\) denote the smallest normal subgroup M with G/M locally indicable. If X is a 2- complex, \(G=\pi_ 1(X),\) and \(N\triangleleft G\) a normal subgroup whose covering complex \(\tilde X\) satisfies \(H_ 2(\tilde X;{\mathbb{F}}_ p)=0\) for all p, then G/r(N) is of cohomology dimension \(\leq 2\). This applies, e.g., in the situation when X is a subcomplex of a 2-complex Y and \(N=\ker(\pi_ 1(X)\to \pi_ 2(Y)).\) And it yields substantial restriction on the centre of G. 2) A G-module M is said to be perfect if \({\mathbb{Z}}\otimes_{{\mathbb{Z}}G}M=0.\) If G is a Ḏ(R)-group then, clearly, G cannot have any non-trivial projective perfect \({\mathbb{Z}}G\)-module. Using this remark and the results above the authors show: If G has a transfinite descending subnormal series \(\{G_{\alpha}\}\) with \(\cap G_{\alpha}=1\) and each \(G_{\alpha}/G_{\alpha +1} R_{\alpha}\)- indicable for some \(R_{\alpha}\). Then G admits no non-trivial projective perfect G-module. Corollary: Let C be a connected n-complex with fundamental group \(\pi_ 1(X)\) of cohomology dimension \(\leq n\) and \(\pi_ i(X)=0\), \(2\leq i<n\). If the Hurewicz map \(\pi_ n (X) \to H_ n (X)\) is trivial and \(G\) has a subnormal series as above then \(X\) is aspherical.
zero divisor and unit problem in group rings
projective perfect \({\mathbb{Z}}G\)-module
scientific article; zbMATH DE number 3845837
scientific article