On approximation properties of semidirect products of groups

Arzhantseva, Goulnara; Gal, Światosław R.

Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 125-130.

Soit $ℛ$ une classe de groupes fermée par rapport aux extensions (scindées) avec un noyau fini et par rapport aux groupes multi-résiduellement $ℛ$. Nous montrons que $ℛ$ contient toutes les extensions (scindées) de type $\{$finiment engendré résiduellement fini$\}$–par–$ℛ$. Nous obtenons en corollaire qu’une extension scindée avec un noyau finiment engendré résiduellement fini et un quotient surjonctif est surjonctive. Cela restait inconnu, même pour les produits directs d’un groupe surjonctif avec les entiers $\mathbf{Z}$.

Let $ℛ$ be a class of groups closed under taking (split) extensions with finite kernel and fully residually $ℛ$–groups. We prove that $ℛ$ contains all (split) $\{$finitely generated residually finite $\}$–by–$ℛ$ groups. It follows that a split extension with a finitely generated residually finite kernel and a surjunctive quotient is surjunctive. This remained unknown even for direct products of a surjunctive group with the integers $\mathbf{Z}$.

Classification : 20E26, 20E22, 20E25, 37B05, 37B10
Mots clés : Residually finite groups, surjunctive and sofic groups, semidirect product

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