Combinatorial cost: A coarse setting
Author:
Tom Kaiser
Journal:
Trans. Amer. Math. Soc. 372 (2019), 2855-2874
MSC (2010):
Primary 05C25
DOI:
https://doi.org/10.1090/tran/7716
Published electronically:
May 7, 2019
MathSciNet review:
3988596
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Abstract | References | Similar Articles | Additional Information
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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Additional Information
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