Title: Combinatorial cost: a coarse setting,
Abstract: The main inspiration for this paper is a paper by Elek where he introduces
combinatorial cost for graph sequences. We show that having cost equal to 1 and
hyperfiniteness are coarse invariants. We also show `cost-1' for box spaces
behaves multiplicatively when taking subgroups. We show that graph sequences
coming from Farber sequences of a group have property A if and only if the
group is amenable. The same is true for hyperfiniteness. This generalises a
theorem by Elek. Furthermore we optimise this result when Farber sequences are
replaced by sofic approximations. In doing so we introduce a new concept:
property almost-A.