Minimal \(H_3\) actions and simple quotients of a discrete 6-dimensional nilpotent group (Q1288523)

There is exactly one connected, simply connected, nilpotent, 3-dimensional Lie group, namely the Heisenberg group \(G_3\). In higher dimensions analogous groups exist, but are not unique in dimensions 5 and above. The group \(G_3\) is naturally equal to \(\mathbb{R}^3\) as a set, and has a co-compact lattice subgroup \(H_3\). In earlier joint papers with S. Walters, the author has studied the groups of dimensions 4 and 5, respectively. Here similar problems are studied for dimension 6. The main concentration is on a particular group \(G_{6,4}\) and, in particular, its lattice subgroup \(H_{6,4}\). Various presentations of \(H_{6,4}\) are given, and its \(C^*\)-algebra is studied. The various infinite dimensional simple quotients of \(C^*(H_{6,4})\) are identified. Use is made of the notion of a \(C^*\)-crossed product to make some of these descriptions in terms of lower dimensional objects.