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For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$.
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The wreath product of ℤ \mathbb{Z} with ℤ \mathbb{Z} has Hilbert compression exponent 2 3 \frac{2}{3}
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