Finite flat spaces

TL;DR

This paper characterizes finite metric spaces that can be almost isometrically embedded into various classes of flat spaces, revealing equivalences among these classes and implications for Markov type constants.

Contribution

It establishes that multiple classes of flat spaces share the same almost isometric embeddability conditions, introducing the concept of finite flat spaces and their properties.

Findings

Almost isometric embeddability conditions are equivalent for several classes of flat spaces.

Connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

The paper introduces the notion of finite flat spaces and explores their synthetic definitions.

Abstract

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less then $1 + \epsilon$. We show that almost isometric embeddability conditions are equal for following classes of spaces (a) Quotients of Euclidean spaces by isometric actions of finite groups, (b) $L_2$-Wasserstein spaces over Euclidean spaces, (c) Compact flat manifolds, (d) Compact flat orbifolds, (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 this conditions finite flat spaces. The question about synthetic definition naturally arises. 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 $1$.