The notion of almost finiteness for group actions on compact spaces is an analogue of both hyperfiniteness in the setting of measure-preserving actions and of Z-stability in the setting of C*-algebras and is related to dynamical comparison in a way that is reminiscent of the link between Z-stability and strict comparison in the Toms-Winter conjecture. I will explain how this relationship with dynamical comparison can be illuminated by means of the small boundary property, leading to new classification results for crossed product C*-algebras.
This is joint work with Gabor Szabo.
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