Hyperlinear and sofic groups: a brief guide

Unrestricted wreath products and sofic 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.

Sofic metric groups and continuous logic

We define sofic, weakly sofic, linear sofic and hyperlinear metric groups and discuss some issues involving axiomatizability of these classes in continuous logic.

Examples of hyperlinear groups without factorization property

In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg's factorization property. This group provides also an example of a sofic Kazhdan group which is not residually finite, answering a question of Elek and Szabo. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in an assertion of Kirchberg and provide an example of a group which does not have the factorization property and is still a subgro…

Metric Approximations of Wreath Products

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 general metric approximations of groups under wreath products. Our main result is the following. Suppose that $H$ is a sofic group and $G$ is a countable, discrete group. If $G$ is sofic, hyperlinear, weakly sofic, or linear sofic, then $G\wr H$ is also sofic, hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant metric ap…

群的言语积与言语圈积的永久性性质

Permanence properties of verbal products and verbal wreath products of groups

本文研究群的言语积与言语圈积，证明sofic性、Haagerup性质等多项性质在幂零积等操作下得以保持，并应用于自由群的商。

陪集作用、光环积和群的度量近似

Sofic actions, halo products, and metric approximations of groups

本文提出sofic C-作用的概念，展示其在构造新群类及统一现有结果中的作用。

Quantifying metric approximations of discrete groups

We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group $G$ by a family of groups $\mathcal{F}=\{(G_α, d_α, k_α, \varepsilon_{α})\}_{α\in I, }$ where each group $G_α$ is equipped with a bi-invariant metric $d_α$ and a dimension $k_α$, for strictly positive real numbers $\varepsilon_{α}$ such that $\inf_{α}\varepsilon_{α}>0$. Through the notion of a residually amenable profile that we introduce, our approach generalizes classical isoperimetric (aka Folner) profiles of amenable…

Some closure results for C-approximable groups

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 …

具有性质(T)的群的灵活Hilbert-Schmidt稳定性与超线性性的对比

Flexible Hilbert-Schmidt stability versus hyperlinearity for property (T) groups

本研究探讨性质(T)群的灵活Hilbert-Schmidt稳定性与超线性性的关系，证明灵活HS-稳定下性质(T)群中心扩张的超线性性，给出非超线性群存在的新方向，是相关工作的Hilbert-Schmidt版本。

不变随机子群、Soficity 与 Lück 行列式猜想

Invariant Random Subgroups, Soficity, and Lück's determinant conjecture

本文将 Lück 行列式猜想推广到自由群的不变随机子群，构造出满足该猜想但非余 sofic 的 IRS，并利用非局域博弈与 MIP*=RE 技术研究 von Neumann 代数的 Connes 可嵌入性。

Metric topological groups: their metric approximation and metric ultraproducts

We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every separable topological group with left-invariant metric. In particular, there is a countable sequence of finite groups with left-invariant metric such that every finite subset of an arbitrary topological group with left-invariant metric may be approximated by all but finitely many of them. We compare our results with related concepts such as sofic groups, hype…

Stability, cohomology vanishing, and non-approximable groups

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\mathrm{Sym}(n)$ (in the sofic case) or the finite dimensional unitary groups ${\rm U}(n)$ (in the hyperlinear case)? In the case of ${\rm U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by ${\rm U}(n)$ with respect to the Frobenius norm $\|T\|…

Traces on group C*-algebras, sofic groups and Lück's conjecture

We give an alternate proof of a Theorem of Elek and Szabo establishing Lück's determinant conjecture for sofic groups. Our proof is based on traces on group C*-algebras. We briefly discuss the relation with Atiyah's problem on the integrality of $L^2$-Betti numbers.

高阶Brin--Thompson群具有Kazhdan性质

Higher Brin--Thompson groups are Kazhdan

本文给出代数判据，证明高阶Brin--Thompson群nV在n较大时具有Kazhdan性质(T)，并统一证明nV、Aut(F_n)及EL_n(R)均具有性质(T)，附录还证明nV非超线性且非Schatten p-可近似。