Classification and discrete cocompact subgroups of some 7-dimensional connected nilpotent groups (Q2368572)

For real connected nilpotent groups, \(7\) is the lowest dimension where there are infinitely many non-isomorphic groups, and also where some groups (indeed, uncountably many) have no discrete cocompact subgroups. \textit{T. Skjelbred and T. Sund} [C. R. Acad. Sci., Paris, Sér. A 286, 241--242 (1978; Zbl 0375.17006)] have identified one infinite family \(\mathcal G\) of \(7\)-dimensional groups. Discrete cocompact subgroups \(H\) were identified, for some groups in \(\mathcal G\), by the author in [Kodai Math. J. 25, 209--226 (2002; Zbl 1019.22003)], along with simple quotients of \(C^\ast(H)\) and relevant flows \((H_3, \mathbf{T}^3)\). In this paper, such \(H\) and attributes are determined for more groups in \(\mathcal G\). The first step in this process is to give, in Theorem 2, a quite detailed variant of Skjelbred's and Sund's result, which leads to the identification of all groups in \(\mathcal G\) that admit a discrete cocompact subgroup \(H\) (Theorem 12) and of the infinite dimensional simple quotients of \(C^\ast(H)\) (Theorem 13). These identifications are somewhat not explicitly given, and to make some of them more explicit, and especially to identify the relevant flows, the author considers other ways of classifying the members of \(\mathcal G\). Furthermore, just one of the forms in the Skjelbred and Sund classification (or real groups) is shown to classify the entire family of complex groups analogous to \(\mathcal G\) (Theorem 7).