  We show how to equip the crossed product between a group of polynomial growth
and a compact quantum metric space with a compact quantum metric space
structure. When the quantum metric on the base space arises from a spectral
triple, which is compatible with the action of the group, we furthermore show
that the crossed product becomes a spectral metric space. Lastly, we analyse
the spectral triple at the crossed product level from the point of view of
unbounded $KK$-theory and show that it arises as an internal Kasparov product
of unbounded Kasparov modules.
