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On the quasi-isometric classification of permutational wreath products

Vincent Dumoncel

TL;DR

The article develops a comprehensive quasi-isometric classification for permutational wreath products $F\wr_{G/M}G$, under finite presentation and non-coarse-separation hypotheses, extending prior results on lamplighters. The authors prove an embedding theorem that factors coarse embeddings through a geometric model built from a graph product, and then show how leaf-preserving quasi-isometries induce base-pair quasi-isometries and scaling data, with a detailed dichotomy depending on co-amenability of the base subgroup. In the non-amenable case, quasi-isometries are governed by the prime divisors of lamp groups, while in the amenable case scaling groups come into play, yielding a robust classification that applies to iterated wreath products and halo constructions. These results yield precise rigidity and flexibility phenomena for permutational lamplighters and enable broad applications, including biLipschitz classifications and corollaries for direct-product bases. The work combines embedding theorems, quasi-median geometry, cone-offs, and new descriptions of lamplighter graphs to advance understanding of large-scale geometry in permutational contexts.

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.

On the quasi-isometric classification of permutational wreath products

TL;DR

The article develops a comprehensive quasi-isometric classification for permutational wreath products , under finite presentation and non-coarse-separation hypotheses, extending prior results on lamplighters. The authors prove an embedding theorem that factors coarse embeddings through a geometric model built from a graph product, and then show how leaf-preserving quasi-isometries induce base-pair quasi-isometries and scaling data, with a detailed dichotomy depending on co-amenability of the base subgroup. In the non-amenable case, quasi-isometries are governed by the prime divisors of lamp groups, while in the amenable case scaling groups come into play, yielding a robust classification that applies to iterated wreath products and halo constructions. These results yield precise rigidity and flexibility phenomena for permutational lamplighters and enable broad applications, including biLipschitz classifications and corollaries for direct-product bases. The work combines embedding theorems, quasi-median geometry, cone-offs, and new descriptions of lamplighter graphs to advance understanding of large-scale geometry in permutational contexts.

Abstract

In this article, we initiate the study of the large-scale geometry of permutational wreath products of the form , where is finitely presented and where is a normal subgroup of 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 , and are quasi-isometric if and only if and 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.
Paper Structure (25 sections, 57 theorems, 199 equations)

This paper contains 25 sections, 57 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Coarse geometry
  5. Quasi-isometries of pairs
  6. Quasi-median geometry
  7. The embedding theorem
  8. The approximation group
  9. A geometric model
  10. Proof of the embedding theorem
  11. Large-scale geometry of permutational lamplighters
  12. An alternative description of permutational lamplighters
  13. Leaf-preserving, aptolic and non-aptolic quasi-isometries
  14. Preserving zones
  15. Coning-off graphs
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $F_{1}, F_{2}$ be non-trivial finite groups and $H_{1}, H_{2}$ be two finitely presented groups. Suppose that $H_{1}$ is one-ended.

Theorems & Definitions (124)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Proposition 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Proposition 1.10
  • Theorem 1.11
  • ...and 114 more