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The Haagerup property is not invariant under quasi-isometry

Mathieu Carette, Kevin Whyte, Sylvain Arnt, Thibault Pillon, Alain Valette

Abstract

Using the work of Cornulier-Valette and Whyte, we show that neither the Haagerup property nor weak amenability is invariant under quasi-isometry of finitely generated groups.

The Haagerup property is not invariant under quasi-isometry

Abstract

Using the work of Cornulier-Valette and Whyte, we show that neither the Haagerup property nor weak amenability is invariant under quasi-isometry of finitely generated groups.

Paper Structure

This paper contains 7 sections, 11 theorems, 5 equations.

Table of Contents

  1. Definitions
  2. Generalized Baumslag-Solitar groups
  3. The role of locally compact groups
  4. Introduction
  5. Holonomy and Model Spaces
  6. Folded holonomy and Lifting
  7. Building the Quasi-isometry

Key Result

Theorem 2

There exists two finitely generated groups $\Gamma$, $\Lambda$ which are quasi-isometric such that $\Gamma$ has the Haagerup property and is weakly amenable and $\Lambda$ has neither of those properties.

Theorems & Definitions (28)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Theorem 4: Whyte10
  • Remark 5
  • Theorem 6: CornulierValette12
  • proof : Proof of Theorem \ref{['theorem:main']}
  • Remark 7
  • Remark 8
  • Remark 9
  • ...and 18 more