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FINITE FLAT SPACES

Part of: Metric geometry

Published online by Cambridge University Press:  14 August 2019

Vladimir Zolotov
Vladimir Zolotov*
Affiliation:
Steklov Institute of Mathematics, Russian Academy of Sciences, 27 Fontanka, 191023 St. Petersburg, Russia University of Cologne, Albertus-Magnus-Platz, 50923 Köln, Germany Mathematics and Mechanics Faculty, St. Petersburg State University, Universitetsky pr., 28, Stary Peterhof, 198504, Russia email [email protected]
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Abstract

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.

  1. (a) Quotients of Euclidean spaces by isometric actions of finite groups.

  2. (b) $L_{2}$-Wasserstein spaces over Euclidean spaces.

  3. (c) Compact flat manifolds.

  4. (d) Compact flat orbifolds.

  5. (e) Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.)

We call spaces which satisfy these conditions finite flat spaces. Since Markov-type constants depend only on finite subsets, we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

MSC classification

Primary: 51F99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 1010 - 1017
Copyright
Copyright © University College London 2019 

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