On $L^1$-approximation of groups

TL;DR

This paper addresses a fundamental open problem in group theory and operator algebras by proving that there exist groups not approximable in the Schatten 1-norm, resolving a question left open for the case p=1.

Contribution

It proves the existence of groups that are not approximated in the Schatten 1-norm, completing the understanding of group approximation properties across all p-values.

Findings

Established the existence of non-approximable groups in Schatten 1-norm.

Resolved the open problem for the case p=1 in group approximation theory.

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.