Almost commuting matrices and stability for product groups

We prove that any product of two non-abelian free groups, =F₌₊, for m, k 2, is not Hilbert–Schmidt stable. This means that there exist asymptotic representations ₍ U (dₙ) 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.