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Lamplighter-like geometry of groups

Anthony Genevois, Romain Tessera

Abstract

In this article, we introduce halo products as a natural generalisation of wreath products. They also encompass lampshuffler groups $\mathrm{FSym}(H) \rtimes H$ and lampcloner groups $\mathrm{FGL}(H) \rtimes H$, as well as many possible variations based for instance on braid groups, Thompson's groups, mapping class groups, automorphisms of free groups. We build a geometric framework that allows us to study the large-scale geometry of halo groups, providing refined invariants distinguishing various halo groups up to quasi-isometry.

Lamplighter-like geometry of groups

Abstract

In this article, we introduce halo products as a natural generalisation of wreath products. They also encompass lampshuffler groups and lampcloner groups , as well as many possible variations based for instance on braid groups, Thompson's groups, mapping class groups, automorphisms of free groups. We build a geometric framework that allows us to study the large-scale geometry of halo groups, providing refined invariants distinguishing various halo groups up to quasi-isometry.
Paper Structure (42 sections, 93 theorems, 235 equations, 1 figure)

This paper contains 42 sections, 93 theorems, 235 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Embedding Theorem.
  3. Algebraic consequences.
  4. First geometric consequences.
  5. Aptolic quasi-isometries.
  6. Growth of lamps.
  7. About the proof of the Embedding Theorem.
  8. Organisation of the article.
  9. Halos of groups
  10. Lampshufflers and lampjugglers.
  11. Lampdesigner.
  12. Lampcloner.
  13. Verbal wreath products.
  14. Others.
  15. Non-examples.
  16. ...and 27 more sections

Key Result

Theorem 1.2

Let $\mathscr{L}H$ be a finitely generated halo product with $L(H)$ locally finite. Let $Z$ be a geodesic metric space satisfying the thick bigon property. For every coarse embedding $Z \to \mathscr{L}H$, the image of $Z$ lies in a neighbourhood of some pseudo-leaf.

Figures (1)

  • Figure 1: Configuration from the proof of Proposition \ref{['prop:SquareLeaves']}.

Theorems & Definitions (224)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Definition 1.10
  • ...and 214 more