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Limits of finite homogeneous metric spaces. (English) Zbl 1328.54020

From the author’s introduction: Itai Benjamini asked me if the spheres \(S^2\), and in general which manifolds, can be approximated by finite homogeneous metric spaces.
We will say that a complete metric space is approximable (or can be approximated) by finite homogeneous metric spaces if it is a limit of such in the Gromov-Hausdorff topology.
Theorem 1.1. A metric space \(X\) is approximable by finite homogeneous metric spaces if and only if it admits a compact group of isometries \(G\) which acts transitively and whose identity connected component \(G^\circ\) is Abelian.
Corollary 1.2. If \(X\) is approximable by finite homogeneous metric spaces then \(X\) is compact, the connected components of \(X\) are inverse limits of tori, and the quotient space of connected components \(X/\sim\) is a transitive totally disconnected space hence is either finite or homeomorphic to the Cantor set.
We will see that a connected component of \(X\) is an inverse limit of tori in the strong, group theoretic, manner: it is homeomorphic to \(\varprojlim T_n\), where the \(T_n\) are compact Abelian Lie groups, and the associated maps are surjective homomorphisms.
In particular, Corollary 1.3. The only manifolds that can be approximated by finite homogenous metric spaces are tori.

MathOverflow Questions:

Scaling limits for groups

MSC:

54E35 Metric spaces, metrizability
54E50 Complete metric spaces
57N99 Topological manifolds
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