Unrestricted wreath products and sofic groups

Arzhantseva, Goulnara; Berlai, Federico; Finn-Sell, Martin; Glebsky, Lev

doi:10.1142/s021819671950005x

Cited by 5 publications

(6 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It remains open for sofic and hyperlinear groups, cf. [1,Question 4.1]. In a particular instance when the kernel is a finite cyclic group, a positive answer to Question 8 would imply the surjunctivity of Deligne's non-residually finite central extensions [6], of its SL n analogues [11] and of its recent arithmetic analogues [9].…”

Section: Question 8 Let G Be a Non-split Extension With A Finite Kermentioning

confidence: 99%

On approximation properties of semidirect products of groups

Arzhantseva¹,

Gal²

2020

Annales Mathématiques Blaise Pascal

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Question 8 Let G Be a Non-split Extension With A Finite Kermentioning

confidence: 99%

On approximation properties of semidirect products of groups

Arzhantseva¹,

Gal²

2020

Annales Mathématiques Blaise Pascal

Self Cite

View full text Add to dashboard Cite

show abstract

“…where the second equality is due to Lemma 2.15 (2). Moreover, since φ and q are surjective, Φ is surjective.…”

Section: By Means Of Theorem 211 It Is Easy To Deduce Thatmentioning

confidence: 91%

Permanence properties of verbal products and verbal wreath products of groups

Brude,

Sasyk

2019

Preprint

View full text Add to dashboard Cite

By means of analyzing the notion of verbal products of groups, we show that soficity, hyperlinearity, amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking k-nilpotent products of groups, while being orderable is not. We also study these properties for solvable and Burnside products of groups. We then show that if two discrete groups are sofic, (respectively have the Haagerup property), their restricted verbal wreath product arising from nilpotent, solvable and certain Burnside products is also sofic (resp. has the Haagerup Property). We also prove related results for hyperlinear, linear sofic and weak sofic approximations. Finally, we give applications combining our work with the Shmelkin embedding to show that certain quotients of free groups are sofic or have the Haagerup property.

show abstract

“…Definition. Let G be a group with a weight function δ and let K be a group with a bi-invariant metric d. Given F ⊆ G, ε > 0, and a map φ : G → K such that φ(1) = 1, we say that (1) φ is (F, ε, d)-multiplicative if d(φ(g)φ(g ′ ), φ(gg ′ )) < ε for all g, g ′ ∈ F ; (2) φ is (F, δ, d)-injective if d(φ(g), 1) ≥ δ(g) for all g ∈ F \ {1};…”

Section: Metric Approximation In Groupsmentioning

confidence: 99%

Metric approximations of unrestricted wreath products when the acting group is amenable

Brude

Sasyk

2021

Communications in Algebra

View full text Add to dashboard Cite

We give a simple and unified proof showing that the unrestricted wreath product of a weakly sofic, sofic, linear sofic, or hyperlinear group by an amenable group is weakly sofic, sofic, linear sofic, or hyperlinear, respectively. By means of the Kaloujnine-Krasner theorem, this implies that group extensions with amenable quotients preserve the four aforementioned metric approximation properties. We also discuss the case of co-amenable groups. introductionGiven two groups G and H, their unrestricted wreath product G ≀≀ H is, by definition, the semidirect product ( H G) ⋊ θ H, where H acts on the direct product H G by shifting coordinates as follows: θ h x h h∈H := x h h h∈H , for x h h∈H ∈ H G. The purpose of this article is to provide a simple and unified proof of the following statement.1.1. Theorem. Let H be an amenable group and let G be a group.(1By means of the Kaloujnine-Krasner theorem, [12], and by keeping in mind that these four metric approximation properties pass to subgroups and are preserved under taking direct products, it is easy to see that Theorem 1.1 and the next result are equivalent. Corollary (Extension Theorem).Let G be a group with a normal subgroup N such that the quotient G/N is amenable. If N is weakly sofic, sofic, linear sofic, or hyperlinear; then G is weakly sofic, sofic, linear sofic, or hyperlinear, respectively.The sofic and hyperlinear cases of Theorem 1.1 have been proved by Arzhantseva, Berlai, and Glebsky in [2], where they also gave the application of the Kaloujnine-Krasner theorem mentioned above. Some of the extension results are older. Indeed, the sofic one is due to Elek and Szabo in [7] and the linear sofic one is due to Arzhantseva and Păunescu in [3]. More recently, in [11], Holt and Rees showed that certain metric approximations on groups, including weakly sofic, are preserved under taking extensions with amenable quotients. In a slightly different direction, in [10], Hayes and Sale proved that the restricted wreath product of a group G having one of the metric approximation properties listed in Theorem 1.1 by an acting sofic group, preserves the approximation property of G.In [2, §4.3] the authors explained why their techniques could not deal with the weakly sofic case of Theorem 1.1. The motivation of the present article was to see whether ideas we used in [4, §5], some of which can be traced back to [10,11], would serve to give a direct proof of this fact, without requiring the result on extensions of [11]. Here we achieve this in a self-contained manner that also allows us to

show abstract

Unrestricted wreath products and sofic groups

Cited by 5 publications

References 12 publications

On approximation properties of semidirect products of groups

On approximation properties of semidirect products of groups

Permanence properties of verbal products and verbal wreath products of groups

Metric approximations of unrestricted wreath products when the acting group is amenable

Contact Info

Product

Resources

About