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Weak property $(\mathrm{T}_{L^p})$ for discrete groups

Emilie Mai Elkiær

Abstract

We show that, for a countable discrete group $Γ$, property $(\mathrm{T}_{L^p})$ of Bader, Furman, Gelander and Monod is equivalent to the property that, whenever an $L^p$-representation of $Γ$ admits a net of almost invariant unit vectors, it has a non-zero invariant vector. Central in the proof is to show that the closure of the group of $\mathbb{T}$-valued $1$-coboundaries is a sufficient criteria for strong ergodicity of ergodic p.m.p. actions.

Weak property $(\mathrm{T}_{L^p})$ for discrete groups

Abstract

We show that, for a countable discrete group , property of Bader, Furman, Gelander and Monod is equivalent to the property that, whenever an -representation of admits a net of almost invariant unit vectors, it has a non-zero invariant vector. Central in the proof is to show that the closure of the group of -valued -coboundaries is a sufficient criteria for strong ergodicity of ergodic p.m.p. actions.
Paper Structure (11 sections, 21 theorems, 36 equations)

This paper contains 11 sections, 21 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Polish groups
  5. Actions on measure spaces
  6. Ergodicity and strong ergodicity
  7. The $\mathbb{T}$-valued measurable functions as a topological group
  8. Groups of $1$-cocycles and $1$-coboundaries
  9. Proof of Theorem \ref{['introthm:B1_closed_implies_strongly_ergodic']}
  10. Proof of Theorem \ref{['introthm:weakTLp_and_TLp']}
  11. On the possibility of an easier proof

Key Result

Theorem C

A countable discrete group has property $(\mathrm{T}_{L^p})$ if and only if it has weak property $(\mathrm{T}_{L^p})$.

Theorems & Definitions (36)

  • Definition A: Property $(\mathrm{T}_{\mathcal{E}})$
  • Definition B: Weak property $(\mathrm{T}_{\mathcal{E}})$
  • Theorem C: Theorem \ref{['thm:weakTLp_and_TLp']}
  • Theorem D: Theorem \ref{['thm:B1_closed_implies_strongly_ergodic']}
  • Theorem E: Theorem \ref{['thm:Lp0_is_not_an_Lp-space']}
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3: Open Mapping Theorem for Polish groups
  • Theorem 2.4: Connes--Weiss
  • Remark 2.5
  • ...and 26 more