K-theory for C*-algebras associated to lattices in Heisenberg Lie groups

We present methods for computing the K-groups of a variety of C*-algebras associated to lattices in Heisenberg Lie groups, including twisted. group C*-algebras and Azumaya algebras over the corresponding nilmanifolds. A precise formula for the rank of the above K-groups is given, and it is shown that any twisted group C*-algebra over such a lattice Gamma is KK-equivalent to an ordinary group C*-algebra corresponding to a possibly different lattice Gamma(0). We also give applications of our methods to the calculation of K-groups for certain twisted transformation group C*-algebras and certain continuous trace algebras whose spectra are tori.