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

Summary 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. In 21] one infinite family <InlineEquation ID=IE"1"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"2"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"3"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"4"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"5"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"6"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"7"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"8"><EquationSource Format="TEX"><!CDATA$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathcal{G}$ of 7-dimensional groups was identified and classified. Discrete cocompact subgroups H were identified for some groups in $\mathcal{G}$ in 10], along with simple quotients of $C^{*}(\mathrm{H})$ and relevant flows $(\mathrm{H}_3,\mathbf{T}^3)$. In this paper, such H and attributes are determined for more groups in $\mathcal{G}$; in particular, the members of $\mathcal{G}$ that admit discrete cocompact subgroups are identified precisely. In achieving some of these results, we consider other known ways of classifying the groups in $\mathcal{G}$, and also the classification of the analogous family of complex groups.