Nonlinear type and metric embeddings of lamplighter spaces

TL;DR

This paper explores the geometric and metric properties of lamplighter spaces over metric spaces, establishing equivalences among several properties and characterizing spaces where the traveling salesman problem is efficiently solvable.

Contribution

It characterizes lamplighter spaces with finite Nagata dimension, Markov type 2, and other properties, linking them to efficient TSP solutions and embeddings into products of trees.

Findings

Lamplighter spaces with certain properties are equivalent.

Characterization of spaces where TSP can be solved efficiently.

Embedding results into products of $ ext{R}$-trees for spaces with biLipschitz embeddings into $ ext{R}^n$.

Abstract

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.