Minimal H₃ actions and simple quotients of discrete 7-dimensional nilpotent groups

Abstract

For connected nilpotent groups, 7 is the lowest dimension where there are infinitely many non-isomorphic groups, and also where some groups have no discrete cocompact subgroups. Here one infinite family of 7-dimensional connected groups is studied, discrete cocompact subgroups H are found for some of them, and then the faithful simple quotients A of C^*(H) are identified. Such A are shown to be isomorphic to C^*-crossed products C^*(H₃, \mathscr{C}(T³)) generated by some intriguing effective minimal distal flows (H₃, T³), where H₃ is the discrete 3-dimensional Heisenberg group.