渐近一致复杂性与顺从性

一致不可顺从性、代价和第一个l^2-贝蒂数

Uniform non-amenability, cost, and the first l^2-Betti number

本文证明了对于任意可数群$\G$，$2β_1(\G)\leq h(\G)$是唯一的，其中$β_1(\G)$是第一个$\ell^2$ -贝蒂数，$h(\G)$是一致等周常数.特别地，具有非零第一$\ell^2$ -贝蒂数的可数群是一致不顺从的。然后，我们定义等周常数的框架内测量的等价关系。对于类型为$\IIi$的遍历测量等价关系$R$，$R$的一致等周常数$h(R)$在轨道等价下是不变的，并且满足$$ 2β_1(R)\leq 2C(R)-2\leq h(R), $$，其中$β_1(\R)$是第一个$\ell^2$ -贝蒂数，$C(R)$是$R$在Levitt意义下的代价（特别地，$h(R)$是非平凡不变量）。与群的情况相反，$\IIi$类型的一致不服从测量的等价关系总是包含不服从的子树。一致等周常数$h(\G)$的遍历版本$h_e(\G)$被定义为与$α$相关联的等价关系$R_α$的一致等周常数$h(\R_α)$的$\G$的所有基本自由遍历和测度保持动作$α$上的下确界。通过建立与保测度等价关系代价的联系，证明了对于实秩至少为2的半单李群中的任意格$\G$，$h_e(\G)=0$（而$h_e(\G)$一般不为零）.

拓扑顺从性的一个同调刻画

A homological characterization of topological amenability

推广Block和Weinberger关于顺从性的刻画，引入了紧空间上群作用的一致有限同调的概念，并利用它给出了作用拓扑顺从性的同调刻画.通过考虑$G$在其Stone- \vCech紧化上的自然作用的情形，得到了该群正合性的一个同调刻画，回答了Nigel Higson的一个问题.

粗不顺从性与小特征值覆盖

Coarse non-amenability and covers with small eigenvalues

给定一个闭黎曼流形M和一个从M的基本群到秩为2的自由群的满射，我们构造了M的一个有限单正则覆盖塔{M_n}_{n=0}^{ \infty }，使得Laplacian的第一非零特征值λ_1（M_n）在n趋于无穷大时收敛于零.这是第一个例子，这样的塔是不能获得一致的准等距（或甚至一致的粗等价）由以前已知的方法，其中的基本群的M是应该surject到一个顺从的群体。

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。

Skew-amenability of topological groups

We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of thi…

Finite-dimensional approximation properties for uniform Roe algebras

We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.

Amenability versus property (T) for non locally compact topological groups

For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property (T) (i.e. admits a finite Kazhdan set). If an amenable topological group with property (T) admits a faithful …

On equivariant asymptotic dimension

The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension", and "$d$-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell--Jones and Borel conjectures. We prove that groups of null equivariant asymptotic dimension are exactly virtually cyclic groups. We show that a covering of the boundary always extends to a covering of the whole compactification. We provide a number of characterisations of equivariant asymptot…

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…

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.

渐近维数与一致嵌入

Asymptotic dimension and uniform embeddings

我们证明了有限渐近维数空间的型函数估计其Hilbert（或任何$l^p$）压缩。该方法允许获得压缩的lampighter组$Z\wr Z$，它具有无穷渐近维数的下界。

局部紧阿贝尔群的一致渐近上密度

On uniform asymptotic upper density in locally compact abelian groups

从N，Z^d，R^d或σ-有限阿贝尔群的最经典情形出发，本文定义了一般局部紧阿贝尔群的渐近一致上密度的概念.即使有点令人惊讶，新的概念被证明是经典情况Z^d，R^d的正确推广。新的概念是用来扩展一些类似的结果，以前只获得了经典的情况下或西格玛有限的阿贝尔群。特别地，我们证明了Furstenberg关于Z的一个著名结果的如下推广：如果在一般的局部紧阿贝尔群G中，G的子集S有正的一致渐近上密度，则S-S是并合的.