Hyperbolic Metric Spaces and Stochastic Embeddings

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 $\ell^1$.