On the quasi-isometric classification of permutational wreath products

In this article, we initiate the study of the large-scale geometry of permutational wreath products of the form $F{\wr }_{H\mathrm{/}N}H$, where $H$ is finitely presented and where $N$ is a normal subgroup of $H$ satisfying a certain assumption of non coarse separation. The main result is a complete classification of such permutational wreath products up to quasi-isometry, building up on previous works from Genevois and Tessera. For instance, we show that, for $d\ge k\ge 2$, ${\mathbb{Z}}_n{\wr }_{{\mathbb{Z}}^k}{\mathbb{Z}}^d$ and ${\mathbb{Z}}_m{\wr }_{{\mathbb{Z}}^k}{\mathbb{Z}}^d$ are quasi-isometric if and only if $n$ and $m$ are powers of a common number. We also discuss biLipschitz equivalences between permutational wreath products, their scaling groups, as well as the quasi-isometric classification of other halo products built out of such permutational lamplighters.