Non p-norm approximated Groups

TL;DR

This paper extends previous results showing certain finitely presented groups cannot be approximated by almost-homomorphisms to unitary groups under Frobenius norms to a broader class of Schatten-p-norms for all p between 1 and infinity.

Contribution

It proves that the non-approximability result holds for all Schatten-p-norms with 1<p<∞, generalizing earlier findings for Frobenius norms.

Findings

Non-approximability extends to all Schatten-p-norms for 1<p<∞

Finitely presented groups cannot be approximated by almost-homomorphisms under these norms

Addresses a question posed in an ICM lecture about generalizing previous results

Abstract

It was shown in a previous work of the first named author with De Chiffre, Glebsky and Thom that there exists a finitely presented group which cannot be approximated by almost-homomorphisms to the unitary groups $U(n)$ equipped with the Frobenius norms (a.k.a as $L^2$ norm, or the Schatten-2-norm). In his ICM18 lecture, Andreas Thom asks if this result can be extended to general Schatten-p-norms. We show that this is indeed the case for $1<p< \infty$.