Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/proc/14526
Published electronically: July 30, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B85, 05C12, 20F65, 30L05

Retrieve articles in all journals with MSC (2010): 46B85, 05C12, 20F65, 30L05


Additional Information

Mikhail I. Ostrovskii
Affiliation: Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, New York 11439
Email: ostrovsm@stjohns.edu

Beata Randrianantoanina
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: randrib@miamioh.edu

DOI: https://doi.org/10.1090/proc/14526
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