  We show that any free action of a connected Lie group of polynomial growth on
a finite dimensional locally compact space has finite tube dimension. This is
shown to imply that the associated crossed product C*-algebra has finite
nuclear dimension. As an application we show that C*-algebras associated with
certain aperiodic point sets in connected Lie groups of polynomial growth are
classifiable. Examples include cut-and-project sets constructed from
irreducible lattices in products of connected nilpotent Lie groups.
