Discrete cocompact subgroups of the four-dimensional nilpotent connected Lie group and their group C*-algebras

Let G(4) be the unique, connected, simply connected, four-dimensional, nilp otent Lie group. In this paper, the discrete cocompact subgroups H of G(4) are classified and shown to be in 1-1 correspondence with triples (p(1), p( 1), p(3)) is an element of Z(3) that satisfy p(2), p(3) > 0 and a certain r estriction on p(1), The K-groups of the group C*-algebra C*(H) are computed and shown to involve all three parameters. Furthermore, for each such subg roup H, the set of faithful simple quotients (i.e., those generated by a fa ithful representation of 1-1) of the group C*-algebra C*(H) is shown to be independent of p(1) and p(3) and to be in 1-1 correspondence with the irrat ional theta '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) a re shown to be isomorphic to matrix algebras over irrational rotation algeb ras. (C) 2001 Academic Press.