Symbolic substitution systems beyond abelian groups

Symbolic substitution systems are an important source of aperiodic Delone sets in abelian locally compact groups. In this article, we consider a large class of non-abelian nilpotent Lie groups with dilation structures, which we refer to as rationally homogeneous Lie groups with rational spectrum (RAHOGRASPs). We show, by explicit construction, that every RAHOGRASP admits a lattice with a primitive symbolic substitution system and hence contains a weakly aperiodic linearly repetitive Delone set. These are the first examples of weakly aperiodic linearly repetitive Delone sets in non-abelian Lie groups. Building on our previous work, we establish unique ergodicity of the corresponding Delone dynamical systems. Our construction applies in particular to all two-step nilpotent Lie groups defined over $\mathbb Q$ such as the Heisenberg group. In this case, the Delone sets in question are in fact strongly aperiodic, i.e. the underlying action of the corresponding Delone dynamical system is free.
Comment: 51 pages