## 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.## References

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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