UNIFORM PROPERTY Γ AND THE SMALL BOUNDARY PROPERTY

We prove that, for a free action α: G X of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property Γ of the Cartan subalgebra (C(X) ⊆ C(X) α G). The reverse implication has been demonstrated by Kerr and Szabó, from which we obtain that these two conditions are equivalent. We moreover show that, if α is also minimal, then almost finiteness of α is implied by tracial Z-stability of the subalgebra (C(X) ⊆ C(X) α G). The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if α: G X and β: H Y are free actions and α has the small boundary property, then α×β: G×H X ×Y has the small boundary property. An analogous permanence property is obtained for almost finiteness in case α and β are free minimal actions.