Discrete Cocompact Subgroups of the Four-Dimensional Nilpotent Connected Lie Group and Their Group C∗-Algebras1

Let G4 be the unique, connected, simply connected, four-dimensional, nilpotent Lie group. In this paper, the discrete cocompact subgroups H of G4 are classified and shown to be in 1–1 correspondence with triples p1 p2 p3 ∈ 3 that satisfy p2 p3 > 0 and a certain restriction on p1. The K-groups of the group C∗-algebra C∗ H are computed and shown to involve all three parameters. Furthermore, for each such subgroup H, the set of faithful simple quotients (i.e., those generated by a faithful representation of H) of the group C∗-algebra C∗ H is shown to be independent of p1 and p3 and to be in 1–1 correspondence with the irrational θ’s in 0 1/2 . The other infinite-dimensional simple quotients of C∗ H (those generated by a representation of H that is not faithful) are shown to be isomorphic to matrix algebras over irrational rotation algebras