On approximation properties of semidirect products of groups

Abstract

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 Z. Sur les propriétés d'approximation des produits semi-directs des groupes Résumé 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 Z.

On dual surjunctivity and applications

“…Using a result of Arzhantseva and Gal [2], we can now obtain the same closure property that they have originally obtained for the class of surjunctive groups.…”

“…Item (1) has been proved in Corollary 3.9, and in order to prove (2), it suffices to show that virtually dual surjunctive groups are dual surjunctive. We proceed as in [2, Lemma 6].…”

“…Proof. We need (2). If (1) is satisfied, i.e., G is polycyclic-by-finite, then by [12, Theorem 1.2], the assumption on completely positive entropy implies that the set .X / is dense in X; that is, (1) implies (2).…”

“…We need (2). If (1) is satisfied, i.e., G is polycyclic-by-finite, then by [12, Theorem 1.2], the assumption on completely positive entropy implies that the set .X / is dense in X; that is, (1) implies (2). Notice also that each equivalence class in is a translate of .X /, so actually each equivalence class is dense.…”