  The aim of the paper is to answer the following question: does
$\kappa$-deformation fit into the framework of noncommutative geometry in the
sense of spectral triples? Using a compactification of time, we get a discrete
version of $\kappa$-Minkowski deformation via $C^*$-algebras of groups. The
dynamical system of the underlying groups (including some Baumslag--Solitar
groups) is 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.
