Quantum metrics on crossed products with groups of polynomial growth

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper K"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">KK</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -theory and show that it arises as an internal Kasparov product of unbounded Kasparov modules.