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A characterization of superreflexivity through embeddings of lamplighter groups

Authors: Mikhail I. Ostrovskii and Beata Randrianantoanina
Journal: Proc. Amer. Math. Soc. 147 (2019), 4745-4755
MSC (2010): Primary 46B85; Secondary 05C12, 20F65, 30L05
Published electronically: July 30, 2019
MathSciNet review: 4011509
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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

Beata Randrianantoanina
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
MR Author ID: 333168

Keywords: Distortion of a bilipschitz embedding, horocyclic product of trees, lamplighter group, Lipschitz map, metric embedding, Ribe program, superreflexivity, word metric
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
Article copyright: © Copyright 2019 American Mathematical Society