Nonlinear type and metric embeddings of lamplighter spaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918418929680384 |
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| author | Gartland, C. Randrianantoanina, B. Randrianarivony, N. L. |
| author_facet | Gartland, C. Randrianantoanina, B. Randrianarivony, N. L. |
| contents | We prove that for all metric spaces $X$ the following properties of the lamplighter space $\mathsf{La}(X)$ are equivalent:
(1) $\mathsf{La}(X)$ has finite Nagata dimension, (2) $\mathsf{La}(X)$ has Markov type 2, (3) $\mathsf{La}(X)$ does not contain the Hamming cubes with uniformly bounded biLipschitz distortion,
(4) $\mathsf{La}(X)$ admits a weak biLipschitz embedding into a finite product of $\mathbb{R}$-trees.
We characterize metric spaces $X$ for which $\mathsf{La}(X)$ satisfies properties (1)-(4) as those whose traveling salesman problem can be solved ``as efficiently" as the traveling salesman problem in $\mathbb{R}$. We also prove that if such metric spaces $X$ admit a biLipschitz embedding into $\mathbb{R}^n$, then
$\mathsf{La}(X)$ admits a biLipschitz embedding into the product of $3n$ $\mathbb{R}$-trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29011 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlinear type and metric embeddings of lamplighter spaces Gartland, C. Randrianantoanina, B. Randrianarivony, N. L. Functional Analysis We prove that for all metric spaces $X$ the following properties of the lamplighter space $\mathsf{La}(X)$ are equivalent: (1) $\mathsf{La}(X)$ has finite Nagata dimension, (2) $\mathsf{La}(X)$ has Markov type 2, (3) $\mathsf{La}(X)$ does not contain the Hamming cubes with uniformly bounded biLipschitz distortion, (4) $\mathsf{La}(X)$ admits a weak biLipschitz embedding into a finite product of $\mathbb{R}$-trees. We characterize metric spaces $X$ for which $\mathsf{La}(X)$ satisfies properties (1)-(4) as those whose traveling salesman problem can be solved ``as efficiently" as the traveling salesman problem in $\mathbb{R}$. We also prove that if such metric spaces $X$ admit a biLipschitz embedding into $\mathbb{R}^n$, then $\mathsf{La}(X)$ admits a biLipschitz embedding into the product of $3n$ $\mathbb{R}$-trees. |
| title | Nonlinear type and metric embeddings of lamplighter spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2603.29011 |