Homological aperiodic tilings of 3-dimensional geometries
  We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds appearing in the geometrization conjecture. 
Piotr W. Nowak and Shmuel Weinberger
math.GR math.DS math.GT math.MG