Joint work with Joan Bosa, Francesc Perera and Jianchao Wu

Motivated by the Toms-Winter conjecture and Kerr’s notion of almost finiteness for actions of amenable discrete groups on compact metric spaces, which is regarded as a dynamical analogue of $\mathcal{Z}$-stability in this setting, we propose a generalisation of almost finiteness to actions of discrete groups on general C*-algebras which we coin almost elementariness. Our starting point is a generalisation of Kerr's notion of a castle which we define as a simultaneous approximation of the algebra and the action, up to an arbitrarily small remainder in a dynamically tracial sense, involving a dynamical version of the Cuntz semigroup. It turns out that various different natural smallness conditions are all equivalent. In the case of no group action our condition is a weak form of tracial AF-ness or tracial nuclear dimension 0. We can show that in this case almost elementariness is equivalent to $\mathcal{Z}$-stability for separable simple nuclear algebras, thus it maybe added as another equivalent condition to the Toms-Winter conjecture. Moreover, almost elementary actions lead to $\mathcal{Z}$-stable crossed products, in line with it being a kind of dynamical $\mathcal{Z}$-stability.