Minimal H_3 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 $C^{*}(\roman H)$ are identified. Such A are shown to be isomorphic to $C^{*}$-crossed products $C^{*}(\roman H_ 3,\Cal C(\Bbb T^3))$ generated by some intriguing effective minimal distal flows $(\roman H_3,\Bbb T^ 3)$, where $\roman H_3$ is the discrete 3-dimensional Heisenberg group.