Stability for product groups and property $(τ)$

Abstract: We study the notion of permutation stability (or P-stability) for countable groups. Our main result provides a wide class of non-amenable product groups which are not P-stable. This class includes the product group $\Sigma\times\Lambda$, whenever $\Sigma$ admits a non-abelian free quotient and $\Lambda$ admits an infinite cyclic quotient. In particular, we obtain that the groups $\mathbb F_m\times\mathbb Z^d$ and $\mathbb F_m\times\mathbb F_n$ are not P-stable, for any integers $m,n\geq 2$ and $d\geq 1$. This implies that P-stability is not closed under the direct product construction, which answers a question of Becker, Lubotzky and Thom. The proof of our main result relies on a construction of asymptotic homomorphisms from $\Sigma\times\Lambda$ to finite symmetric groups starting from sequences of finite index subgroups in $\Sigma$ and $\Lambda$ with and without property $(\tau)$. Our method is sufficiently robust to show that the groups covered are not even flexibly P-stable, thus giving the first such non-amenable residually finite examples.