关于词范数意义下度量弱 sofic 的群

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…

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。

Ultraproducts, weak equivalence and sofic entropy

In this work, we study pmp actions of countable groups on arbitrary diffuse probability spaces under the point of view of weak equivalence. We will show that any such an action is weakly equivalent to an action on a standard probability space. We also propose a metric on the space of actions modulo weak equivalence which is equivalent to the topology of Abért and Elek. We will give a simpler proof of the compactness of the space, showing that convergence is characterized by ultraproducts. Using this topology, we will show that a profinite action is weakly equivalent to an ultraproduct of finit…

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。

顺从代数的线性 sofic 表示

Linear sofic representations of amenable algebras

本文研究代数的线性 sofic 近似概念，证明对于无零因子的有限生成顺从 K-代数，所有线性 sofic 表示共轭，并应用于秩度量下的弱稳定性问题。

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 remarks on finitarily approximable groups

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length fun…

作用的不可近似性与Kazhdan性质(T)

Inapproximability of actions and Kazhdan's property (T)

该研究针对具有Kazhdan性质(T)的可数群G与有限生成群D，通过sofic逼近的扩张子图性质，推导D的LEF性，明确非LEF时G×D作用的极限性质，并给出几乎自同构的自改进定理作为关键技术支撑。

Weakly surjunctive groups and symbolic group varieties

In this paper, we introduce the classes of weakly surjunctive and linearly surjunctive groups which include all sofic groups and more generally all surjunctive groups. We investigate various properties of such groups and establish in particular a reversibility and invertibility theorem for injective endomorphisms of symbolic group varieties over weakly surjunctive group universes with algebraic group alphabets in arbitrary characteristic. We also obtain novel evidence related to Kaplansky's stable finiteness conjecture.

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…

Weak amenability of CAT(0) cubical groups

We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations for which the quantities z^{l(g)} are matrix coefficients. Here l is a length function on G obtained from the combinatorial distance function on the complex X.

Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy

We prove a max-min theorem for weak containment in the context of algebraic actions. Namely, we show that given an algebraic action of $G$ on $X,$ there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is $G\curvearrowright X/Y$-ergodic "in the presence of $G\curvearrowright X$". We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal …