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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A characterization of superreflexivity through embeddings of lamplighter groups
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by Mikhail I. Ostrovskii and Beata Randrianantoanina PDF
Proc. Amer. Math. Soc. 147 (2019), 4745-4755 Request permission

Abstract:

We prove that finite lamplighter groups $\{\mathbb {Z}_2\wr \mathbb {Z}_n\}_{n\ge 2}$ with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space and therefore form a set of test spaces for superreflexivity. Our proof is inspired by the well-known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover $\mathbb {Z}_2\wr \mathbb {Z}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings.
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Additional Information
  • Mikhail I. Ostrovskii
  • Affiliation: Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, New York 11439
  • MR Author ID: 211545
  • Email: ostrovsm@stjohns.edu
  • Beata Randrianantoanina
  • Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
  • MR Author ID: 333168
  • Email: randrib@miamioh.edu
  • Received by editor(s): July 17, 2018
  • Received by editor(s) in revised form: December 17, 2018
  • Published electronically: July 30, 2019
  • Additional Notes: The first author was supported by the National Science Foundation under Grant Number DMS–1700176.
  • Communicated by: Stephen Dilworth
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4745-4755
  • MSC (2010): Primary 46B85; Secondary 05C12, 20F65, 30L05
  • DOI: https://doi.org/10.1090/proc/14526
  • MathSciNet review: 4011509