Complexity of non-abelian cut-and-project sets of polytopal type I: special homogeneous Lie groups

The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like $r^{{\mathrm {homdim}}(G) \dim (H)}$ . Further, we generalize the concept of acceptance domains to locally compact second countable groups.