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Hyperbolic Metric Spaces and Stochastic Embeddings

Chris Gartland

TL;DR

The work develops a comprehensive framework connecting stochastic biLipschitz embeddings into tree-like spaces with isomorphisms of Lipschitz free spaces to $L^1$-spaces. By exploiting Bonk–Schramm hyperbolic fillings and boundary/bulk-embedding principles, the authors establish that infinite proper metric spaces stochastically embedding into $\mathbb{R}$-trees yield $LF(X)\approx L^1$, while snowflakes of finite Nagata-dimension spaces admit stochastic embeddings into ultrametrics with $LF(X)\approx \ell^1$. They further show that the Lipschitz free space of hyperbolic $n$-space coincides (up to isomorphism) with that of Euclidean $n$-space, and that infinite finitely generated hyperbolic groups stochastically embed into $\mathbb{R}$-trees, yielding $LF(\Gamma)\approx \ell^1$ and admitting proper, uniformly Lipschitz affine actions on $\ell^1$. The results combine deep geometric-analytic methods with Lipschitz-free-space techniques to produce new $L^1$-embeddability phenomena for large-scale spaces and provide tools for analyzing group actions on Banach spaces with sharp geometric flavor.

Abstract

Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim towards applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into $\mathbb{R}$-trees have Lipschitz free spaces isomorphic to $L^1$-spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into $\mathbb{R}$-trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) Every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to $\ell^1$. (2) The Lipschitz free space over hyperbolic $n$-space is isomorphic to the Lipschitz free space over Euclidean $n$-space. (3) Every infinite, finitely generated hyperbolic group stochastically embeds into an $\mathbb{R}$-tree, has Lipschitz free space isomorphic to $\ell^1$, and admits a proper, uniformly Lipschitz affine action on $\ell^1$.

Hyperbolic Metric Spaces and Stochastic Embeddings

TL;DR

The work develops a comprehensive framework connecting stochastic biLipschitz embeddings into tree-like spaces with isomorphisms of Lipschitz free spaces to -spaces. By exploiting Bonk–Schramm hyperbolic fillings and boundary/bulk-embedding principles, the authors establish that infinite proper metric spaces stochastically embedding into -trees yield , while snowflakes of finite Nagata-dimension spaces admit stochastic embeddings into ultrametrics with . They further show that the Lipschitz free space of hyperbolic -space coincides (up to isomorphism) with that of Euclidean -space, and that infinite finitely generated hyperbolic groups stochastically embed into -trees, yielding and admitting proper, uniformly Lipschitz affine actions on . The results combine deep geometric-analytic methods with Lipschitz-free-space techniques to produce new -embeddability phenomena for large-scale spaces and provide tools for analyzing group actions on Banach spaces with sharp geometric flavor.

Abstract

Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim towards applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into -trees have Lipschitz free spaces isomorphic to -spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into -trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) Every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to . (2) The Lipschitz free space over hyperbolic -space is isomorphic to the Lipschitz free space over Euclidean -space. (3) Every infinite, finitely generated hyperbolic group stochastically embeds into an -tree, has Lipschitz free space isomorphic to , and admits a proper, uniformly Lipschitz affine action on .
Paper Structure (21 sections, 42 theorems, 116 equations)

This paper contains 21 sections, 42 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Main Results, Outline of their Proofs, and Structure of Article
  3. Relationship to Existing Work
  4. Preliminaries
  5. Metric Spaces and Maps between them
  6. Banach and Bochner-Lebesgue Spaces
  7. Lipschitz Free Spaces
  8. Nagata Dimension and Lipschitz Extensions
  9. Hyperbolic Metric Spaces
  10. Ultrametric Spaces and $\mathbb{R}$-Trees
  11. Stochastic Embeddings and Lipschitz Free Spaces
  12. Random Maps and Stochastic Embeddings
  13. Spaces of Random Lipschitz Functions
  14. Stochastic Embeddings into $\mathbb{R}$-Trees and Isomorphisms to $L^1$-Spaces
  15. Stochastic Embeddings into Ultrametric Spaces
  16. ...and 6 more sections

Key Result

Theorem A

Let $X$ be an infinite, proper, pointed metric space. If $X$ stochastic biLipschitzly embeds into a pointed $\mathbb{R}$-tree, then $\mathop{\mathrm{LF}}\nolimits(X) \approx L^1$ or $\mathop{\mathrm{LF}}\nolimits(X) \approx \ell^1$, with the latter isomorphism holding if and only if the completion o

Theorems & Definitions (85)

  • Theorem A: Theorem \ref{['thm:stochasticembedtreeLFisoL1']}
  • Theorem B
  • proof
  • Theorem C: Theorem \ref{['thm:Hypstochasticums']}
  • Theorem D: Corollary \ref{['cor:LF(Hn)']}
  • Theorem E: Theorem \ref{['thm:l1action']}
  • Lemma 2.1: almost Theorem C, AGPP
  • proof
  • Theorem 2.2
  • proof
  • ...and 75 more