  We construct discrete versions of $\kappa$-Minkowski space related to a
certain compactness of the time coordinate. We show that these models fit into
the framework of noncommutative geometry in the sense of spectral triples. The
dynamical system of the underlying discrete groups (which include some
Baumslag--Solitar groups) is heavily used in order to construct \emph{finitely
summable} spectral triples. This allows to bypass an obstruction to
finite-summability appearing when using the common regular representation. The
dimension of these spectral triples is unrelated to the number of coordinates
defining the $\kappa$-deformed Minkowski spaces.
