Frobenius Non-Stability of Nilpotent Groups

TL;DR

This paper demonstrates that finitely generated non-virtually cyclic nilpotent groups are not Frobenius stable, extending the result to Schatten p-norms, highlighting limitations in approximating group representations.

Contribution

It establishes the non-stability of a broad class of nilpotent groups under Frobenius and Schatten p-norms, advancing understanding of group stability properties.

Findings

Finitely generated non-virtually cyclic nilpotent groups are not Frobenius stable.

The non-stability result extends to Schatten p-norms for 1<p≤∞.

The proof provides a unified approach for these norm-based stability analyses.

Abstract

A countable discrete group is said to be Frobenius stable if every function from the group to unitary matrices that is "almost multiplicative" in the Frobenius norm is "close" to a unitary representation in the Frobenius norm. The purpose of this paper is to show that finitely generated nilpotent groups that are not virtually cyclic are not Frobenius stable. Our argument proves the same result for other unnormalized Schatten $p$-norms with $1<p\le\infty$.