Almost commuting matrices and stability for product groups

Abstract

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