Strong almost finiteness

A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on the boundary of a tile is uniformly small. We give various equivalents for strong almost finiteness. In particular, we prove that Property A together with the F\o lner property is equivalent to strong almost finiteness. Using these characterizations, we show that graphs of subexponential growth and Schreier graphs of amenable groups are always strongly almost finite, generalizing the celebrated result of Downarowicz, Huczek and Zhang about amenable Cayley graphs, based on graph theoretic rather than group theoretic principles. We give various equivalents to Property A for graphs, and show that if a sequence of graphs of Property A (in a uniform sense) converges to a graph $G$ in the neighborhood distance (a purely combinatorial analogue of the classical Benjamini-Schramm distance), then their Laplacian spectra converge to the Laplacian spectrum of $G$ in the Hausdorff distance. We apply the previous theory to construct a new and rich class of classifiable $C^{\star}$-algebras. Namely, we show that for any minimal strong almost finite graph $G$ there are naturally associated simple, nuclear, stably finite $C^{\star}$-algebras that are classifiable by their Elliott invariants.