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Distortion of embeddings of binary trees into diamond graphs


Authors: Siu Lam Leung, Sarah Nelson, Sofiya Ostrovska and Mikhail Ostrovskii
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 46B85; Secondary 05C12, 30L05
DOI: https://doi.org/10.1090/proc/13750
Published electronically: August 30, 2017
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Abstract: Diamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other; for example superreflexivity of Banach spaces can be characterized both in terms of binary trees (Bourgain, 1986) and diamond graphs (Johnson-Schechtman, 2009). In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. This question was answered in the negative by Ostrovskii (2014), who left it open to determine the order of growth of the distortions. The main purpose of this paper is to get a sharp up-to-a-logarithmic-factor estimate for the distortions of embeddings of binary trees into diamond graphs and, more generally, into diamond graphs of any finite branching $ k\ge 2$. Estimates for distortions of embeddings of diamonds into infinitely branching diamonds are also obtained.


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Additional Information

Siu Lam Leung
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: sleung1@kent.edu

Sarah Nelson
Affiliation: Department of Mathematics and Statistics, Hunter College, CUNY, New York, New York 10065
Email: sarah.nelson07@yahoo.com

Sofiya Ostrovska
Affiliation: Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey
Email: sofia.ostrovska@atilim.edu.tr

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

DOI: https://doi.org/10.1090/proc/13750
Keywords: Binary tree, diamond graph, distortion of a bilipschitz embedding, Lipschitz map
Received by editor(s): August 8, 2016
Received by editor(s) in revised form: March 28, 2017
Published electronically: August 30, 2017
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society