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arXiv · 1711.00413

Combinatorial cost: a coarse setting

Tom Kaiser
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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.

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Tom Kaiser. 2017-11-01. Combinatorial cost: a coarse setting. https://arxiv.org/abs/1711.00413

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