Almost commuting matrices and stability for product groups

TL;DR

The paper demonstrates that certain product groups are not stable under Hilbert-Schmidt norms, showing that almost commuting matrices can be far from truly commuting ones, answering a longstanding question in matrix theory.

Contribution

It proves the non-stability of product groups of free groups under Hilbert-Schmidt norms and constructs counterexamples to almost commuting matrices.

Findings

Product groups of free groups are not Hilbert-Schmidt stable.

Existence of matrices that almost commute but are far from truly commuting matrices.

Negative resolution of a 1969 question by Rosenthal on almost commuting matrices.

Abstract

We prove that any product of two non-abelian free groups, $\Gamma=\mathbb F_m\times\mathbb F_k$, for $m,k\geq 2$, is not Hilbert-Schmidt stable. This means that there exist asymptotic representations $\pi_n:\Gamma\rightarrow \text{U}({d_n})$ with respect to the normalized Hilbert-Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$, with respect to the normalized Hilbert-Schmidt norm, but $A,B$ are not close to any matrices $A',B'$ such that $A'$ commutes with $B'$ and $B'^*$. This settles in the negative a natural version of a question concerning almost commuting matrices posed by Rosenthal in 1969.