  In this paper we generalize several results on separated nets in Euclidean
space to separated nets in connected simply connected nilpotent Lie groups. We
show that every such group $G$ contains separated nets that are not biLipschitz
equivalent. We define a class of separated nets in these groups arising from a
generalization of the cut-and-project quasi-crystal construction and show that
generically any such separated net is bounded displacement equivalent to a
separated net of constant covolume. In addition, we use a generalization of the
Laczkovich criterion to provide `exotic' perturbations of such separated nets.
