Buildings have finite asymptotic dimension
Jan Dymara, Thomas Schick
在本文中，我们证明了任何建筑物的渐近维数是有限的，等于该建筑物中的公寓的渐近维数。
In this note, we show that the asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.
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Preserving Coarse Properties
本文的目的是研究度量空间$X$上由等距作用于有限群$G$的商映射$X\to X/\sim$的粗糙$n$ -to-1函数的保性和共保性.我们主要感兴趣的粗糙性质与渐近维数及其推广有关：具有有限渐近维数，渐近性质C，直有限分解复杂性，可数渐近维数和度量稀疏化性质。我们提供了另一种描述的渐近性质C，我们证明了，一类空间的直有限分解复杂性与可数渐近维数的空间。
Cohomological approach to asymptotic dimension
在有界上同调的基础上引入了渐近上同调的概念，定义了度量空间的上同调渐近维数$\as_{\Z} X$.证明了当渐近维数$\as X$为有限时，它与渐近维数一致.然后利用这一事实构造了一个渐近维数有限的有界几何度量空间$X$的例子，其中$\as(X\times\R)=\as X$。特别地，对于这个例子，由Roe粗上同调定义的粗渐近维数严格地小于其渐近维数。
We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD). The context of all theorems is a finitely generated group $G$ with a word metric and a coarse quasi-action on a metric space $X$. We assume that the quasi-stabilizers have a property $P_1$, and $X$ has the same or sometimes a weaker property $P_2$. Then $G$ also has property $P_2$. We show some sample applications, discuss constraints to further g…
We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition complexity. Moreover, finite APC-decomposition complexity implies property A for metric spaces. We also show that finite APC-decomposition complexity is preserved by direct products of groups and spaces, amalgamated products of groups, and group extensions, among other constructions.
Assouad-Nagata dimension of locally finite groups and asymptotic cones
本文研究了关于Assouad-Nagata维数的两个问题：1）是否存在非零Assouad-Nagata维数的度量空间，使得它的所有渐近锥面都是零维数的？（Dydak and Higes）2）设$G$是一个具有适当左不变度量$d_G$的局部有限群.如果是$\dim_{AN}(G, d_G)>0$，那么$\dim_{AN} (G, d_G)$是无限的吗？（Brodskiy，Dydak和Lang）第一个问题不仅对一般度量空间而且对具有适当左不变度量的离散群都得到了肯定的回答。第二个问题有一个否定解。证明了对每个$n$，存在一个局部有限的Assouad-Nagata维数为$n$的群.给出了任意渐近维数可数群的一个推广
In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincaré inequality. We show that at almost every point $x$ outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at $x$. We also show that, at co-dimension $1$ Hausdorff measure almost every measure-theoretic boundary point of a set $E$ of finite perimeter, there is an asymptotic limit set $(E)_\infty$ corresponding …
We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counter-example was known. We construct a finitely presentable (eve…
Finite asymptotic dimension for CAT(0) cube complexes
本文证明了有限维CAT（0）立方复形的渐近维数在维数上有界。为了实现这一点，我们证明了一个控制着色定理的复杂。我们还证明了每一个CAT（0）立方体复形都是无限维立方体的压缩收缩。作为维数定理的一个例子，我们得到了小消去群的渐近维数的界。
Finite asymptotic dimension and the coarse assembly map
该研究针对具有弱转移的强粗同调理论与弱有限同伦渐近维数的有界粗空间，给出粗装配映射为幻影等价这一结论的简单论证。
Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $Ω_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the rescaling sequence $\left(X, \frac{1}{r_n}d\right).$ The set of all pretangent spaces $Ω_{\infty, \tilde r}^{X}$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X, \tilde r}, ρ_{X})$ whose maximal cliques coincide with $Ω_{\infty, \tilde r}^{X}$ and the weight $ρ_{X}$ is defined by metrics on $Ω_{\infty,…
A characterization for asymptotic dimension growth
我们给出了渐近维数增长的一个刻画。我们将其应用于有限维CAT（0）立方复形，给出了N的另一种证明。Wright关于它们的有限渐近维数的结果。我们还将我们的新的特征有限秩的测地粗中值空间，并建立他们有次指数渐近维数增长。这加强了J.Spakula和N.赖特
We prove that for all $k,m,n \in \mathbb N \cup \{\infty\}$ with $4 \leq k \leq m \leq n$, there exists a finitely generated group $G$ with a finitely generated subgroup $H$ such that the asymptotic dimension of $G$ is $k$, the Assouad-Nagata dimension of $G$ is $m$, and the Assouad-Nagata dimension of $H$ is $n$. This simultaneously answers two open questions in asymptotic dimension theory.
Finite decomposition complexity and asymptotic dimension growth are two generalizations of M. Gromov's asymptotic dimension which can be used to prove property A for large classes of finitely generated groups of infinite asymptotic dimension. In this paper, we introduce the notion of decomposition complexity growth, which is a quasi-isometry invariant generalizing both finite decomposition complexity and dimension growth. We show that subexponential decomposition complexity growth implies property A, and is preserved by certain group and metric constructions.
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