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On $L^1$-approximation of groups

Benjamin Bachner, Alon Dogon, Alexander Lubotzky

TL;DR

The paper settles the $p=1$ case of the Schatten-$p$-norm approximation problem by showing that $ ext{Deligne-type}$ groups, which have a central element $J$ of order $2$ in $\ker_{RF}$, cannot be $\mathscr{G}_p$-approximated when $p \le q$ given $\mathscr{G}_q$-stability. The authors introduce a two-norm strategy, combining different matrix norms on $U(n)$, to deduce inapproximability without requiring full $\mathscr{G}$-stability results in all cases. The main result yields explicit finitely presented groups not $\mathscr{G}_1$-approximated and highlights a path toward MF negation via operator-HS-stability; it also establishes monotonicity properties and connects to broader questions about soficity and hyperlinearity. Overall, the work advances understanding of the limits of finite-dimensional approximations of groups and provides new techniques for proving non-approximability in the Schatten-$p$ framework.

Abstract

A longstanding open problem in the intersection of group theory and operator algebras is whether all groups are MF, that is, approximated by asymptotic representations with respect to the operator norm. More generally, for $1 \leq p \leq \infty$, it has been asked by Thom in his ICM address whether there exist groups which are not approximated with respect to the Schatten $p$-norm. The cases of $1 < p < \infty$ were addressed in previous works. We settle the case $p=1$, solving a question left open by Lubotzky and Oppenheim.

On $L^1$-approximation of groups

TL;DR

The paper settles the case of the Schatten--norm approximation problem by showing that groups, which have a central element of order in , cannot be -approximated when given -stability. The authors introduce a two-norm strategy, combining different matrix norms on , to deduce inapproximability without requiring full -stability results in all cases. The main result yields explicit finitely presented groups not -approximated and highlights a path toward MF negation via operator-HS-stability; it also establishes monotonicity properties and connects to broader questions about soficity and hyperlinearity. Overall, the work advances understanding of the limits of finite-dimensional approximations of groups and provides new techniques for proving non-approximability in the Schatten- framework.

Abstract

A longstanding open problem in the intersection of group theory and operator algebras is whether all groups are MF, that is, approximated by asymptotic representations with respect to the operator norm. More generally, for , it has been asked by Thom in his ICM address whether there exist groups which are not approximated with respect to the Schatten -norm. The cases of were addressed in previous works. We settle the case , solving a question left open by Lubotzky and Oppenheim.

Paper Structure

This paper contains 4 sections, 10 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unitarily invariant matrix norms
  4. Proofs

Key Result

Proposition 1.1

Let $\mathscr{G}$ be a class as above, where $G_n$ are linear groups, and $\Gamma$ a finitely generated group. If $\Gamma$ is $\mathscr{G}$-approximated and $\mathscr{G}$-stable, then it is residually finite.

Theorems & Definitions (17)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Conjecture 1.8
  • Lemma 2.1: Proposition 1.3 in DGLT
  • Lemma 2.2: Proposition 1.4 in DGLT
  • ...and 7 more