Hyperbolic Metric Spaces and Stochastic Embeddings
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 t...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942400118X/type/journal_article |
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| author | Chris Gartland |
| author_facet | Chris Gartland |
| author_sort | Chris Gartland |
| collection | DOAJ |
| description | 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 hyperbolic n-space is isomorphic to the Lipschitz free space over Euclidean n-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
$\ell ^1$
. |
| format | Article |
| id | doaj-art-f429d96eb78143dd894c97ad6699c8e0 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-f429d96eb78143dd894c97ad6699c8e02025-02-07T07:50:22ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.118Hyperbolic Metric Spaces and Stochastic EmbeddingsChris Gartland0Department of Mathematics, University of California San Diego, 9500 Gilman Dr. La Jolla, CA, 92093, USAStochastic 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 hyperbolic n-space is isomorphic to the Lipschitz free space over Euclidean n-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 $\ell ^1$ .https://www.cambridge.org/core/product/identifier/S205050942400118X/type/journal_article51F3020F6730L0546B0346B20 |
| spellingShingle | Chris Gartland Hyperbolic Metric Spaces and Stochastic Embeddings Forum of Mathematics, Sigma 51F30 20F67 30L05 46B03 46B20 |
| title | Hyperbolic Metric Spaces and Stochastic Embeddings |
| title_full | Hyperbolic Metric Spaces and Stochastic Embeddings |
| title_fullStr | Hyperbolic Metric Spaces and Stochastic Embeddings |
| title_full_unstemmed | Hyperbolic Metric Spaces and Stochastic Embeddings |
| title_short | Hyperbolic Metric Spaces and Stochastic Embeddings |
| title_sort | hyperbolic metric spaces and stochastic embeddings |
| topic | 51F30 20F67 30L05 46B03 46B20 |
| url | https://www.cambridge.org/core/product/identifier/S205050942400118X/type/journal_article |
| work_keys_str_mv | AT chrisgartland hyperbolicmetricspacesandstochasticembeddings |