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Hyperlinear approximations to amenable groups come from sofic approximations

Peter Burton, Maksym Chaudkhari, Kate Juschenko, Kyrylo Muliarchyk

Abstract

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The proof is probabilistic, using the concentration of measure in high-dimensional spheres to control the deviation of an operator's matrix coefficients from its trace. As a corollary, we obtain a result connecting stability of sofic approximations with stability of hyperlinear approximations.

Hyperlinear approximations to amenable groups come from sofic approximations

Abstract

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The proof is probabilistic, using the concentration of measure in high-dimensional spheres to control the deviation of an operator's matrix coefficients from its trace. As a corollary, we obtain a result connecting stability of sofic approximations with stability of hyperlinear approximations.
Paper Structure (35 sections, 37 theorems, 172 equations)

This paper contains 35 sections, 37 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Hyperlinear and sofic groups
  3. Statement and discussion of main result
  4. Applications of main result
  5. Stable soficity and hyperlinearity
  6. Free products with amalgamation
  7. Outline of proof of Theorem \ref{['thm.sofic']}
  8. Acknowledgements
  9. Linear algebraic estimates
  10. Orthogonalization bound
  11. Complementary projections and span
  12. Almost commuting unitaries and projections
  13. Analysis on high dimensional spheres
  14. Normalized traces and averages of matrix coefficients
  15. Concentration of measure and deviation bounds
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $G$ be a countable amenable group, let $E$ be a finite subset of $G$ and let $\epsilon > 0$. Then there exists a finite subset $F$ of $G$ and $\delta > 0$ with the following property. If $\alpha:G \to \mathrm{Un}(\mathcal{H})$ is an $(F,\delta)$ hyperlinear approximation to $G$ then there exists

Theorems & Definitions (82)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Definition 1.3
  • Definition 1.4
  • Corollary 1.1
  • Theorem 1.2
  • Lemma 1.1: Main lemma
  • Remark 1.1
  • Proposition 2.1
  • ...and 72 more