Some algebraic and topological properties of the nonabelian tensor product. (Q2843782)

Given two groups \(G\) and \(H\) we assume that each group acts by conjugation on itself (\(^gg':=gg'g^{-1}\)) and that there are suitable actions of each group on the other such that the actions are compatible in the sense: NEWLINE\[NEWLINE^{^gh}g'={^g}(^h(^{g^{-1}}g'))\text{ and }^{^hg}h'={^h}(^g(^{h^{-1}}h'))\text{ for all }g,g'\in G\text{ and }h,h'\in H.NEWLINE\]NEWLINE The nonabelian tensor product \(G\otimes H\) is then defined to be the group generated by the symbols \(g\otimes h\) with the defining relations NEWLINE\[NEWLINEgg'\otimes h=(^gg'\otimes{^g}h)(g\otimes h)\text{ and }g\otimes hh'=(g\otimes h)(^hg\otimes{^h}h')NEWLINE\]NEWLINE [see \textit{R. Brown} et al., J. Algebra 111, 177-202 (1987; Zbl 0626.20038)]. In the special case where \(G=H\) and all actions are conjugation, \(G\otimes G\) is called the nonabelian tensor square. Let \(\mathcal X\) be a class of groups which is closed with respect to taking subgroups and quotient groups. Then we say \(\mathcal X\) is closed with respect to \(\mathrm{NT}\) if \(G\in\mathcal X\) implies that \(G\otimes G\in\mathcal X\). The authors note that in the literature there is a wide range of classes proved to be closed with respect to \(\mathrm{NT}\) including the classes of Chernikov groups, nilpotent groups, soluble groups and polycyclic groups.NEWLINENEWLINE The present paper shows that we also have closure with respect to \(\mathrm{NT}\) for many local classes (such as locally finite, locally nilpotent, etc.), hyper classes (such as hyper-finite, hyper-cyclic, etc.) and classes of groups generalizing finite conjugacy. A central tool in the proof is the following. Suppose that the groups \(G\) and \(H\) are normal subgroups of some group \(M\) (acting compatibly by conjugation on one another). Define the corresponding semidirect product \(\eta(G,H):=((G\otimes H)\rtimes H)\rtimes G\). Now, if \(\mathcal X\) is any class of groups which is closed under taking subgroups and quotient groups and contains both \(G\) and \(H\), then \(G\otimes H\in\mathcal X\).NEWLINENEWLINE The paper ends with a short result on certain topological groups, namely, quasi simply filtered (qsf) groups [see \textit{S. G. Brick} and \textit{M. L. Mihalik}, Math. Z. 220, No. 2, 207-217 (1995; Zbl 0843.57003)] where it is shown that, under suitable hypotheses, if \(G\) and \(H\) are infinite qsf groups, then \(G\otimes H\) is also a qsf group.