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Property FW and wreath products of groups: a simple approach using Schreier graphs

Paul-Henry Leemann, Grégoire Schneeberger

Abstract

The group property FW stands in-between the celebrated Kazdhan's property (T) and Serre's property FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended. It follows from the work of Y. Cornulier that a finitely generated wreath product $G\wr_XH$ has property~FW if and only if both $G$ and $H$ have property FW and $X$ is finite. The aim of this paper is to give an elementary, direct and explicit proof of this fact using Schreier graphs.

Property FW and wreath products of groups: a simple approach using Schreier graphs

Abstract

The group property FW stands in-between the celebrated Kazdhan's property (T) and Serre's property FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended. It follows from the work of Y. Cornulier that a finitely generated wreath product has property~FW if and only if both and have property FW and is finite. The aim of this paper is to give an elementary, direct and explicit proof of this fact using Schreier graphs.

Paper Structure

This paper contains 9 sections, 8 theorems, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Definitions and examples
  4. Ends of Schreier graphs and property FW
  5. Wreath products
  6. Proof of the main result
  7. If $G\wr_XH$ has property FW, then $X$ is infinite.
  8. The group $G$ has property FW.
  9. Acknowledgment

Key Result

Theorem 1.1

Let $G,H$ be two discrete groups with $G$ non-trivial and $X$ a set on which $H$ acts. The wreath product $G \wr_X H$ has property (T) if and only if $G$ and $H$ have property (T) and $X$ is finite.

Figures (5)

  • Figure 1: Fragments of two Cayley graphs of $\mathbf{Z}$ ($2$ ends), for the standard generating set $\{\pm1\}$ and for the generating set $\{\textcolor{red}{\pm2},\textcolor{blue}{\pm3}\}$.
  • Figure 2: Fragments of the Cayley graphs of $\mathbf{Z}^2$ ($1$ end) on the left and of $F_2$ (infinitely many ends) on the right; with standard generating sets.
  • Figure 3: A fragment of a Schreier graph (with $4$ ends) of the free group $F_2=\langle \textcolor{red}{x^{\pm1}},\textcolor{blue}{y^{\pm1}}\rangle$ for the subgroup $H=\{x^2,y^nxy^{-n},xy^nxy^{-n}x^{-1}\ |\ n\in\mathbf{Z}\setminus\{0\}\}$.
  • Figure 4: The path between $x$ and $y$.
  • Figure 5: The leaf structure of the orbital Schreier graph of $G\wr_XH \curvearrowright Y$. Plain edges correspond to generators of the form $(\mathbf 1,t)$ while dotted edges correspond to generators of the form $(\delta_{x_0}^s,1)$.

Theorems & Definitions (16)

  • Theorem 1.1: Cherix2004Neuhauser2005a
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 6 more