Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regular maps from the lamplighter to metabelian groups

Antoine Gournay, Corentin Le Coz

Abstract

We prove that the lamplighter group admits an injective Lipschitz map to any finitely generated metabelian group which is not virtually nilpotent. This implies that finitely generated metabelian groups satisfy the ``analytically thin/analytically thick'' dichotomy recently introduced by Hume, Mackay and Tessera.

Regular maps from the lamplighter to metabelian groups

Abstract

We prove that the lamplighter group admits an injective Lipschitz map to any finitely generated metabelian group which is not virtually nilpotent. This implies that finitely generated metabelian groups satisfy the ``analytically thin/analytically thick'' dichotomy recently introduced by Hume, Mackay and Tessera.
Paper Structure (8 sections, 7 theorems, 19 equations)

This paper contains 8 sections, 7 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Comments and questions
  3. Regular maps
  4. Some simple examples
  5. Metabelian groups
  6. Lamplighter and affine groups
  7. Proof of Theorem \ref{['teomet']}
  8. Separation profile of the lamplighter group

Key Result

Proposition 1.2

Let $G, H$ be graphs with bounded degree such that there exists a regular map $H \to G$. Then, $\mathop{\mathrm{Sep}}\nolimits_H \preceq \mathop{\mathrm{Sep}}\nolimits_G$.

Theorems & Definitions (25)

  • Definition 1.1: Benjamini, Schramm, Timár BST
  • Proposition 1.2: Benjamini, Schramm, Timár BST
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • ...and 15 more