Nonlinear type and metric embeddings of lamplighter spaces

We prove that for all metric spaces $X$ the following properties of the lamplighter space $\mathsf{L}\mathsf{a}(X)$ are equivalent: (1) $\mathsf{L}\mathsf{a}(X)$ has finite Nagata dimension, (2) $\mathsf{L}\mathsf{a}(X)$ has Markov type 2, (3) $\mathsf{L}\mathsf{a}(X)$ does not contain the Hamming cubes with uniformly bounded biLipschitz distortion, (4) $\mathsf{L}\mathsf{a}(X)$ admits a weak biLipschitz embedding into a finite product of $\mathbb{R}$-trees. We characterize metric spaces $X$ for which $\mathsf{L}\mathsf{a}(X)$ 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 $\mathsf{L}\mathsf{a}(X)$ admits a biLipschitz embedding into the product of $3n$ $\mathbb{R}$-trees.