Metric approximations of unrestricted wreath products when the acting group is amenable
Javier Brude, Román Sasyk
备注：
v4: Final version. To appear in Communications in Algebra. A new section was added, which deals with the case when a group has a co-amenable subgroup that is weakly sofic, sofic, linear sofic, or hyperlinear, respectively. 13 pages
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.
We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argument, to hyperlinear-by-amenable groups.
In [BV12] we have proven that, for all hyperbolic groups and for all non-trivial free products $Γ$, the left-right wreath product group $G:=(Z/2Z)^{(Γ)} \rtimes (Γ\times Γ)$ is W$^*$-superrigid. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups $Γ$ having positive first $\ell^2$-Betti number, the same wreath product $G$ is W$^*$-superrigid.
We calculate the exact values of the Følner function $\mathrm{Føl}$ of the lamplighter group $\mathbb{Z}\wr\mathbb{Z}/2\mathbb{Z}$ for the standard generating set. More generally, for any finite group $D$ and $n\geq|D|$, we obtain the exact value of $\mathrm{Føl}(n)$ on the wreath product $\mathbb{Z}\wr D$, for a generating set induced by a generator on $\mathbb{Z}$ and the entire group being taken as generators for $D$. We also describe the Følner sets that give rise to it. Følner functions encode the isoperimetric properties of amenable groups and have previously been studied up to asymptoti…
We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath pr…
We show that any infinite collection $(Γ_n)_{n\in \mathbb N}$ of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic \emph{infinite product rigidity} phenomenon. If $Λ$ is an arbitrary group such that $L(\oplus_{n\in \mathbb N} Γ_n)\cong L(Λ)$ then there exists an infinite direct sum decomposition $Λ=(\oplus_{n \in \mathbb N} Λ_n )\oplus A$ with $A$ icc amenable such that, for all $n\in \mathbb N$, up to amplifications, we have $L(Γ_n) \cong L(Λ_n)$ and $L(\oplus_{k\geq n} Γ_k )\cong L((\oplus_{k\geq n} Λ_k) \oplus A)$. The result is sharp and complements the pre…
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 two C-approximable groups is C-approximable; (ii) the restricted standard wreath product G wr H is C-approximable when G is C-approximable and H is residually finite; and (iii) a group G$ with normal subgroup N is C-approximable when N is C-approximable and G/N is amenable. Our direct product result is valid for LEF, weakly sofic and hyperlinear groups, as …
We consider II$_1$ factors of the form $M=\bar{\bigotimes}_{G}N\rtimes G$, where either i) $N$ is a non-hyperfinite II$_1$ factor and $G$ is an ICC amenable group or ii) $N$ is a weakly rigid II$_1$ factor and $G$ is ICC group and where $G$ acts on $\bar{\bigotimes}_{G}N$ by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting are isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.
We get three types of results on measure equivalence rigidity; direct product groups of Ozawa's class $\mathcal{S}$ groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class $\mathcal{S}$ groups. As consequences, Monod--Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups $A \wr G$, $B \wr Γ$ of non-amenable exact direct product groups $G$, $Γ$ with amenable bases $A$, $B$ are measure equivalent, then $G$ and $Γ$ are measure equivalent. We get Bass--Serr…
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 permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.
In this paper, it is shown that the reduced $\ell^p$-cohomology is trivial for a class of finitely generated amenable groups called transport amenable. These groups are those for which there exist a sequence of measures $ξ_n$ converging to a left-invariant mean and such that the transport cost between $ξ_n$ displaced by multiplication on the right by a fixed element and $ξ_n$ is bounded in $n$. This class contains groups with controlled Følner sequence (such as polycylic groups) as well as some wreath products. Note: Examples 5.1 and 5.2 are incorrect (in particular $\mathbb{Z} \wr \mathbb{Z}$…
We give a positive answer, in the measurable-group-theory context, to von Neumann's problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors.
This article is dedicated to the asymptotic geometry of wreath products $F\wr H := \left( \bigoplus_H F \right) \rtimes H$ where $F$ is a finite group and $H$ a one-ended finitely presented group. Our main result is a complete classification of these groups up to quasi-isometry. More precisely, given two finite groups $F_1,F_2$ and two finitely presented one-ended groups $H_1,H_2$, we show that $F_1 \wr H_1$ and $F_2 \wr H_2$ are quasi-isometric if and only if either (i) $H_1,H_2$ are non-amenable quasi-isometric groups and $|F_1|,|F_2|$ have the same prime divisors, or (ii) $H_1,H_2$ are amen…
We single out a large class of groups ${\mathscr{M}}$ for which the following unique prime factorization result holds: if $Γ_1,\dots,Γ_n\in {\mathscr{M}}$ and $Γ_1\times\dots\timesΓ_n$ is measure equivalent to a product $Λ_1\times\dots\timesΛ_m$ of infinite icc groups, then $n \ge m$, and if $n = m$ then, after permutation of the indices, $Γ_i$ is measure equivalent to $Λ_i$, for all $1\leq i\leq n$. This provides an analogue of Monod and Shalom's theorem \cite{MS02} for groups that belong to ${\mathscr{M}}$. Class ${\mathscr{M}}$ is constructed using groups whose von Neumann algebras admit an…
We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempote…
