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The Local Lifting Property, Property FD, and stability of approximate representations

Francesco Fournier-Facio, Rufus Willett

Abstract

We establish Kirchberg's Local Lifting Property and Lubotzky-Shalom's Property FD for classes of finitely generated groups of central importance in geometric and combinatorial group theory: $3$-manifold groups, limit groups, and certain one-relator groups and right-angled Artin groups. We deduce that such groups are very flexibly stable, with respect to normalized unitarily invariant norms. The exposition is made accessible to operator algebraists and group theorists alike.

The Local Lifting Property, Property FD, and stability of approximate representations

Abstract

We establish Kirchberg's Local Lifting Property and Lubotzky-Shalom's Property FD for classes of finitely generated groups of central importance in geometric and combinatorial group theory: -manifold groups, limit groups, and certain one-relator groups and right-angled Artin groups. We deduce that such groups are very flexibly stable, with respect to normalized unitarily invariant norms. The exposition is made accessible to operator algebraists and group theorists alike.
Paper Structure (34 sections, 70 theorems, 88 equations)

This paper contains 34 sections, 70 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Background on the (L)LP for $C^*$-algebras
  3. Permanence properties for the (L)LP
  4. Group $C^*$-algebras and crossed products
  5. Groups without the (L)LP
  6. Free and amenable groups
  7. Subgroups
  8. Increasing unions
  9. Semidirect products by amenable groups
  10. Free products amalgamated over finite subgroups
  11. Graphs of groups with finite edge groups
  12. More graphs of groups
  13. Central extensions
  14. Amenable extensions
  15. New examples of (L)LP groups
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be LLP and RFD. Let $(\phi_n \colon \Gamma\to U_{k_n})$ be an asymptotic representation of $\Gamma$. Then there exists a sequence of finite-dimensional Hilbert spaces $H_n$, representations $\pi_n \colon \Gamma\to U(H_n)$, and isometric inclusions $v_n \colon \mathbb{C}^{k_n}\to H_n$ su

Theorems & Definitions (201)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 191 more