On the quasi-isometric classification of permutational wreath products

TL;DR

This paper classifies permutational wreath products up to quasi-isometry, revealing how their large-scale geometry depends on the algebraic structure of the groups involved, and extends previous classification results.

Contribution

It provides a complete quasi-isometric classification of permutational wreath products with new criteria based on group parameters, building on prior work.

Findings

$ ext{Z}_n ext{wr}_{ ext{Z}^k} ext{Z}^d$ and $ ext{Z}_m ext{wr}_{ ext{Z}^k} ext{Z}^d$ are quasi-isometric iff $n,m$ are powers of a common number.

Classification depends on the algebraic parameters of the groups involved.

Discusses biLipschitz equivalences and scaling groups of such wreath products.

Abstract

In this article, we initiate the study of the large-scale geometry of permutational wreath products of the form $F\wr_{H/N}H$, 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.