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Uniform local amenability implies Property A


Author: Gábor Elek
Journal: Proc. Amer. Math. Soc. 149 (2021), 2573-2577
MSC (2020): Primary 46L99, 51F99
DOI: https://doi.org/10.1090/proc/15387
Published electronically: March 18, 2021
MathSciNet review: 4246807
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Abstract: In this short note we answer a query of Brodzki, Niblo, Špakula, Willett and Wright [J. Noncommut. Geom. 7 (2013), pp. 583–603] by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser [Trans. Amer. Math. Soc. 372 (2019), pp. 2855–2874] proved that if $\Gamma$ is a finitely generated group and $\{H_i\}^\infty _{i=1}$ is a Farber sequence of finite index subgroups, then the associated Schreier graph sequence is of Property A if and only if the group is amenable. We show however, that there exist a non-amenable group and a nested sequence of finite index subgroups $\{H_i\}^\infty _{i=1}$ such that $\cap H_i=\{e_\Gamma \}$, and the associated Schreier graph sequence is of Property A.


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Additional Information

Gábor Elek
Affiliation: Department of Mathematics And Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom
MR Author ID: 360750
Email: g.elek@lancaster.ac.uk

Received by editor(s): December 10, 2019
Received by editor(s) in revised form: August 24, 2020, and October 9, 2020
Published electronically: March 18, 2021
Additional Notes: The author was partially supported by the ERC Starting Grant “Limits of Structures in Algebra and Combinatorics”.
Communicated by: Patricia Hersh
Article copyright: © Copyright 2021 American Mathematical Society