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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Combinatorial cost: A coarse setting

Author: Tom Kaiser
Journal: Trans. Amer. Math. Soc. 372 (2019), 2855-2874
MSC (2010): Primary 05C25
Published electronically: May 7, 2019
MathSciNet review: 3988596
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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 that “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
Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Emile–Argand 11 2000 Neuchâtel, Switzerland

Received by editor(s): December 14, 2017
Received by editor(s) in revised form: June 14, 2018, August 31, 2018, and September 14, 2018
Published electronically: May 7, 2019
Article copyright: © Copyright 2019 American Mathematical Society