Product groups Σ×Λ are not very flexibly P-stable when Σ admits a non-abelian free quotient and Λ lacks property (τ), so P-stability is not closed under direct products.
Flexible stability and nonsoficity
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
A sofic group $G$ is said to be flexibly stable if every sofic approximation to $G$ can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We establish that if $\mathrm{PSL}_d(\mathbb{Z})$ is flexibly stable for some $d \geq 5$ then there exists a group which is not sofic.
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math.GR 1years
2019 1verdicts
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Stability for product groups and property $(\tau)$
Product groups Σ×Λ are not very flexibly P-stable when Σ admits a non-abelian free quotient and Λ lacks property (τ), so P-stability is not closed under direct products.