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Transactions of the American Mathematical Society

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Measure expanding actions, expanders and warped cones


Author: Federico Vigolo
Journal: Trans. Amer. Math. Soc. 371 (2019), 1951-1979
MSC (2010): Primary 51F99, 05C99, 37A99, 46N99
DOI: https://doi.org/10.1090/tran/7368
Published electronically: August 21, 2018
MathSciNet review: 3894040
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Abstract: We define a way of approximating actions on measure spaces using finite graphs. We then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat unified approach to constructing expanders. We also show that the graphs we obtain are uniformly quasi-isometric to the level sets of warped cones. This way we can also prove non-embeddability results for the latter and restate an old conjecture of Gamburd-Jakobson-Sarnak.


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Additional Information

Federico Vigolo
Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG United Kingdom
Email: vigolo@maths.ox.ac.uk

DOI: https://doi.org/10.1090/tran/7368
Keywords: Approximation, measure, expanders, spectral gap, warped cone.
Received by editor(s): December 16, 2016
Received by editor(s) in revised form: August 1, 2017
Published electronically: August 21, 2018
Additional Notes: This work was funded by the EPSRC Grant 1502483 and the J.T. Hamilton Scholarship. The material is also based upon work supported by the NSF under Grant No. DMS-1440140 while the author was in residence at the MSRI in Berkeley during the Fall 2016 semester.
Article copyright: © Copyright 2018 American Mathematical Society