Combinatorial cost: a coarse setting

TL;DR

This paper explores the concept of combinatorial cost in graph sequences, establishing its invariance properties, its relation to group amenability, and introducing the new notion of property almost-A.

Contribution

It extends the understanding of combinatorial cost as a coarse invariant and generalizes existing theorems to broader classes of graph sequences and group actions.

Findings

Cost equals 1 and hyperfiniteness are coarse invariants.

Cost-1 behaves multiplicatively for subgroups in box spaces.

Graph sequences from Farber sequences have property A iff the group is amenable.

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.