Related papers: Unrestricted wreath products and sofic groups

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…

We define wreath products of cocommutative Hopf algebras, and show that they enjoy a universal property of classifying cleft extensions, analogous to the Kaloujnine-Krasner theorem for groups. We show that the group ring of a wreath product…

Answering some queries of Weiss, we prove that the free product and amenable extensions of sofic groups are sofic as well, and give an example of a finitely generated sofic group that is not residually amenable.

Given sofic approximations for countable, discrete groups $G,H$, we construct a sofic approximation for their wreath product $G\wr H$.

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,…

We show that free products of sofic groups with amalgamation over monotileably amenable subgroups are sofic. Consequently, so are HNN extensions of sofic groups relative to homomorphisms of monotileably amenable subgroups. We also show that…

We investigate closure results for C-approximable groups, for certain classes C of groups with invariant length functions. In particular we prove, each time for certain (but not necessarily the same) classes C that (i) the direct product of…

Given the large class of groups already known to be sofic, there is seemingly a shortfall in results concerning their permanence properties. We address this problem for wreath products, and in particular investigate the behaviour of more…

We prove that graph products of sofic groups are sofic, as are graphs of groups for which vertex groups are sofic and edge groups are amenable.

We characterize the normal extensions of inverse semigroups isomorphic to full restricted semidirect products, and present a Kalouznin-Krasner theorem which holds for a wider class of normal extensions of inverse semigroups than that in the…

Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic…

We define a notion of relative soficity for countable groups with respect to a family of groups. A group is sofic if and only if it is relative sofic with respect to the family consisting only of the trivial group. If a group is relatively…

We develop a theory of soficity for actions on graphs and obtain new applications to the study of sofic groups. We establish various examples, stability and permanence properties of sofic actions on graphs, in particular soficity is…

We prove that all invariant random subgroups of the lamplighter group $L$ are co-sofic. It follows that $L$ is permutation stable, providing an example of an infinitely presented such a group. Our proof applies more generally to all…

Given a morphism $\varphi : G \to A \wr B$ from a finitely presented group $G$ to a wreath product $A \wr B$, we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free…

We introduce the concept of crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system.…

We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open…

In this article we develop a notion of soficity for actions of countable groups on sets. We show two equivalent perspectives, several natural properties and examples. Notable examples include arbitrary actions of both amenable groups and…

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The…

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of…