A note on topological amenability

A note on Shavgulidze's papers concerning the amenability problem for Thompson's group $F$

This paper examines Shavgulidze's postings to the ArXiv and publications which concern his argument that Thompson's group F is amenable. In particular I list specific places where there are errors in each of the postings and publications and give a public clarification of some of the speculation around these papers.

On Shavgulidze's Proof of the Amenability of some Discrete Groups of Homeomorphisms of the Unit Interval

Notes that elaborate on the details of a Theorem of Shavgulidze that implies the amenability of R. J. Thompson's group F.

On amenability and groups of measurable maps

We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],λ)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely amenable. Conversely, we prove that a topological group $G$ is amenable if $L^{0}(G)$ is.

Topological Amenability

Let G be a locally compact topological group and X a compact space with continuous G-action. The main result of this essay states that the following statements are equivalent : 1) The action of G on X is topologically amenable ; 2) Every dual (G, X)-module of type C is a relatively injective Banach G-module ; 3) There is a G-invariant element in the double dual of C(X, L1(G)).

Topological matchings and amenability

We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings. We also show that extremely amenable as well as compactly approximable topological groups satisfy a perfect matching property condition -- the latter even with regard to arbitrary uniform coverings. Finally, we prove that the automorphism group of a Fraïssé limit of a metric Fraïssé class is amenable if and only if the considered metric Fraïssé class has a ce…

关于拓扑全群的顺从性与LEF性质

On Amenability and LEF-ness for topological full groups

本文证明了拓扑自由远端作用的拓扑全群顺从性等价于底群顺从，并推广了LEF性质到剩余有限作用，进而证明自由群上某些Toeplitz子位移的拓扑全群是LEF且sofic。

Amenability of semigroups and common multiples in $\ell^1_+$

In this note, we prove that a semigroup $S$ is left amenable if and only if every two nonzero elements of $\ell^1_+(S)$ have a common nonzero right multiple, where $\ell^1_+(S)$ is the positive part of the Banach algebra $\ell^1(S)$, or equivalently the semiring of finite measures on $S$. This characterization of amenability is new even for groups.

Asymptotic uniform complexity and amenability

We introduce a novel quantity for general dynamical systems, which we call the asymptotic uniform complexity. We prove an inequality relating the asymptotic uniform complexity of a dynamical system to its mean topological matching number. Furthermore, we show that the established inequality yields an exact equation for perfect Hausdorff dynamical systems. Utilizing these results, we conclude that a dynamical system is amenable if its asymptotic uniform complexity equals one, and that the converse is true for perfect Hausdorff dynamical systems, such as non-discrete Hausdorff topological groups…

Strong amenability and the infinite conjugacy class property

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its quotients have the infinite conjugacy class property.

C*-代数酉群的顺从性

Amenability for unitary groups of C*-algebras

本文提出刻画幺正C*-代数酉群顺从性的猜想，证明了其对特定类C*-代数成立，并基于Kirchberg工作给出Ng相关问题的反例。

共轭可 amenability

Conjugacy co-amenability

本文研究了群包含中的一种新性质，即存在非紧支撑的不变态，探讨了其与 proper proximality 的关联。

关于不变均值与预共生子群

On invariant means and pre-syndetic subgroups

本文研究非局部紧拓扑群顺从性变体，证明斜顺从性可被预共生子群继承，并探讨 Malliavin 提出的顺从性变体，建立其 Reiter 型和 Følner 型判据。

Dynamical alternating groups, stability, property Gamma, and inner amenability

We prove that the alternating group of a topologically free action of a countably infinite group $Γ$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $Γ$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Γ$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance o…

可数马尔可夫移位的离散群扩张与压力间隙

Discrete group extensions of countable Markov shifts and pressure gap

本文将关于顺从性和增长间隙的若干经典结果推广到强正递归的可数马尔可夫移位，利用扭曲Patterson--Sullivan测度的方法，并给出扭曲测度的背景及一个例子的估计。

群作用于树的顺从性

Amenability of groups acting on trees

本说明描述了第一个例子的一组是顺从的，但不能得到的子群，从属，扩展和直接限制类的群体局部的次指数增长。它有一个平衡的介绍\[Δ= < b,t|[b,t^2]b^{-1},[[[b,t^{-1}],b],b]>.\]我表明，它的行为传递的3-正则树，和$Γ=< b,b^{t^{-1}}$稳定的顶点，并通过限制的二元有根树。$Γ$是一个“分形群”，由一个三态自动机生成，是黎曼球面上相关覆盖上f（z）=z^2-1迭代的单值作用的离散模拟。$Δ$与Thompson组$F$共享许多属性。主要结果（$Δ$的可修改性）的证明在本形式中是不完整的;请参阅与Balint Virag联合发表的论文arxiv.org/math.GR/0305262，以获得完整的证明。

群在紧空间上的顺从性及相关 Banach 代数

Amenability of group actions on compact spaces and the associated Banach algebras

本文将拓扑群顺从性的经典刻画推广至紧 Hausdorff 空间上的离散群作用，通过引入自然关联的 Banach 代数和弱化的顺从性概念，获得了顺从作用的类似刻画，并证明了加强 Dong 和 Wang (2015) 定理的不动点性质。

On topological groups with an approximate fixed point property

A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that $x_{i}-gx_{i}\longrightarrow 0$ for all $g\in G$. We study the relationship of this property with amenability.

Normed amenability and bounded cohomology over non-Archimedean fields

We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field $K$. To capture the features of classical amenability that induce the vanishing of real bounded cohomology, we introduce the notion of normed $K$-amenability, of which we prove an algebraic characterization. It implies that normed $K$-amenable groups are locally elliptic, and it relates an invariant, the norm of a $K$-amenable group, to the order of its discrete finite $p$-subquotients, where $p$ is the characteristic of the residue field of $K$. Moreover, w…

Amenability, connected components, and definable actions

We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain "weak Bohr compactification" introduced in [24]. Formally, $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjec…

Amenable actions, invariant means and bounded cohomology

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-Čech compactification of the group we obtain a cohomologic…

On Følner sets in topological groups

We extend Følner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and provide an affirmative answer to a question posed by Rosendal.

Amenability and Acyclicity in Bounded Cohomology Theory

Johnson's characterization of amenable groups states that a discrete group $Γ$ is amenable if and only if $H_b^{n \geq 1}(Γ; V) = 0$ for all dual normed $\mathbb{R}[Γ]$-modules V. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective group homomorphism $ϕ\colon Γ\to K$ has amenable kernel H if and only if the induced inflation map $H^\bullet_b(K; V^H) \to H^\bullet_b(Γ; V)$ is an isometric isomorphism for every dual normed $\mathbb{R}[Γ]$-module V. In addition, we obtain an analogous characterization for the (smaller) class o…

A Note on Topologically-Trivial Braids

We give a simple characterization of braids that can be unplaited keeping separately their upper ends and their lower ends tied together

关于阿贝尔拓扑群 Pontryagin 对偶性的若干注记

Some notes on Pontryagin duality of abelian topological groups

本文探讨了阿贝尔pro-Lie群中Pontryagin对偶性相关的若干问题，分析了该领域内的核心方法与主要贡献。