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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 toward 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 hyperbolicn-space is isomorphic to the Lipschitz free space over Euclideann-space and (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$$\ell ^1$$.
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Lipschitz-free Spaces on Finite Metric Spaces
Abstract Main results of the paper are as follows: (1) For any finite metric space $$M$$ the Lipschitz-free space on $$M$$ contains a large well-complemented subspace that is close to $$\ell _{1}^{n}$$ . (2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $$\ell _{1}^{n}$$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.
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- Award ID(s):
- 1700176
- PAR ID:
- 10286174
- Date Published:
- Journal Name:
- Canadian Journal of Mathematics
- Volume:
- 72
- Issue:
- 3
- ISSN:
- 0008-414X
- Page Range / eLocation ID:
- 774 to 804
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4153/S0008414X19000087
