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.
Isoperimetric profiles of lamplighter-like groups
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abstract
Given a finitely generated amenable group $H$ satisfying some mild assumptions, we relate isoperimetric profiles of the lampshuffler group $\mathsf{Shuffler}(H)=\mathsf{FSym}(H)\rtimes H$ to those of $H$. Our results are sharp for all exponential growth groups for which isoperimetric profiles are known, including Brieussel-Zheng groups. This refines previous estimates obtained by Erschler and Zheng and by Saloff-Coste and Zheng. The most difficult part is to find an optimal upper bound, and our strategy consists in finding suitable lamplighter subgraphs in lampshufflers. This novelty applies more generally for many examples of halo products, a class of groups introduced recently by Genevois and Tessera as a natural generalisation of wreath products. Lastly, we also give applications of our estimates on isoperimetric profiles to the existence problem of regular maps between such groups.
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2026 1verdicts
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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.