Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent if and only if p < k/(k+ℓ), via a new notion of orbit equivalence of pairs and stability results for permutational halo products.
Transitivity degrees of countable groups and acylindrical hyperbolicity
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On quantitative orbit equivalence for lamplighter-like groups
Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent if and only if p < k/(k+ℓ), via a new notion of orbit equivalence of pairs and stability results for permutational halo products.