A tracial characterization of Furstenberg's xp, xq conjecture

We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg's xp, xq conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the C*-algebra of the group Z[1/pq] (sic) Z(2). We also compute the primitive ideal space and K-theory of C*( Z[1/pq] (sic) Z(2)).