Almost commuting matrices and stability for product groups

We prove that any product of two non-abelian free groups, Γ=Fₘₖ , for m,k≥ 2 , is not Hilbert–Schmidt stable. This means that there exist asymptotic representations πₙ Γ→ 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.