On the structure of asymptotic expanders
On the structure of asymptotic expanders
In this paper, we study several analytic and graph-theoretic properties of asymptotic expanders. We show that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we use asymptotic expanders to provide uncountably many new counterexamples to the coarse Baum–Connes conjecture. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a C ⁎-algebraic characterisation of expanders for vertex-transitive graphs. We achieve these results by showing that a sequence of asymptotic expanders always admits a “uniform exhaustion by expanders”. This also implies that asymptotic expanders cannot be coarsely embedded into any L p-space.
Asymptotic expanders, Averaging projection, Coarse Baum–Connes conjecture, Følner sets, Quasi-locality, Uniform Roe algebra
Khukhro, Ana
d38850a1-411d-4cd2-9b1c-3f2173b10c50
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5
Vigolo, Federico
914e52c5-0729-42c2-b487-52f2292f1712
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
24 December 2021
Khukhro, Ana
d38850a1-411d-4cd2-9b1c-3f2173b10c50
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5
Vigolo, Federico
914e52c5-0729-42c2-b487-52f2292f1712
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
Khukhro, Ana, Li, Kang, Vigolo, Federico and Zhang, Jiawen
(2021)
On the structure of asymptotic expanders.
Advances in Mathematics, 393, [108073].
(doi:10.1016/j.aim.2021.108073).
Abstract
In this paper, we study several analytic and graph-theoretic properties of asymptotic expanders. We show that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we use asymptotic expanders to provide uncountably many new counterexamples to the coarse Baum–Connes conjecture. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a C ⁎-algebraic characterisation of expanders for vertex-transitive graphs. We achieve these results by showing that a sequence of asymptotic expanders always admits a “uniform exhaustion by expanders”. This also implies that asymptotic expanders cannot be coarsely embedded into any L p-space.
Text
expanderish_graph_structure_Adv_revision_LZV
- Accepted Manuscript
More information
Submitted date: 12 December 2019
Accepted/In Press date: 3 October 2021
e-pub ahead of print date: 16 November 2021
Published date: 24 December 2021
Keywords:
Asymptotic expanders, Averaging projection, Coarse Baum–Connes conjecture, Følner sets, Quasi-locality, Uniform Roe algebra
Identifiers
Local EPrints ID: 452711
URI: http://eprints.soton.ac.uk/id/eprint/452711
ISSN: 0001-8708
PURE UUID: db3ec2d7-0b60-49bc-ad30-7f94ae179192
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Date deposited: 15 Dec 2021 17:34
Last modified: 06 Jun 2024 04:20
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Contributors
Author:
Ana Khukhro
Author:
Kang Li
Author:
Federico Vigolo
Author:
Jiawen Zhang
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