Almost commuting matrices and stability for product groups

Abstract

We prove that any product of two non-abelian free groups, $\Gamma ={\mathbb{F}}_m\times {\mathbb{F}}_k$, for $m,k\ge 2$, is not Hilbert–Schmidt stable. This means that there exist asymptotic representations ${\pi }_n:\Gamma \to \mathrm{U}(d_n)$ 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 $B^{\ast }$, with respect to the normalized Hilbert–Schmidt norm, but $A$, $B$ are not close to any matrices $A^{\prime }$, $B^{\prime }$ such that $A^{\prime }$ commutes with $B^{\prime }$ and $B^{\prime \ast }$. This settles in the negative a natural version of a question concerning almost commuting matrices posed by Rosenthal in 1969.