粗动力学与不动点性质

Bounded Scale Measure and Property A

We introduce a generalization for bounded geometry that we call bounded scale measure. We show that bounded scale measure is a coarse invariant unlike bounded geometry. We then show equivalent definitions for spaces with bounded scale measure and show other properties that spaces with bounded scale measure satisfy. From there, we generalize property A from uniformly discrete metric spaces to large scale spaces with bounded geometry. Lastly, we construct a definition for property A for large scale spaces with bounded scale measure and show that this definition of property A is a coarse invarian…

Coarse free products

We define a notion of free product for coarse spaces that generalizes the corresponding notion of a free product for groups. We show that free products preserve coarse properties such as coarse property C, finite coarse decomposition complexity, and coarse property A. We also give an upper bound estimate on the dimension of a coarse free product in terms of the dimension of its factors.

Asymptotic dimension of coarse spaces via maps to simplicial complexes

It is well-known that a paracompact space $X$ is of covering dimension at most $n$ if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton $K^{(n)}$. We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map $f$ is replaced by variation of $f$ on elements of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces.

Coarse quotients by group actions and the maximal Roe algebra

For a discrete metric space (or more generally a large scale space) $X$ and an action of a group $G$ on $X$ by coarse equivalences, we define a type of coarse quotient space $X_G$, which agrees up to coarse equivalence with the orbit space $X/G$ when $G$ is finite. We then restrict our attention to what we call coarsely discontinuous actions and show that for such actions the group $G$ can be recovered as an appropriately defined automorphism group $\mathsf{Aut}(X/X_G)$ when $X$ satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable gro…

关于Nagata的渐近维数和一个性质

On asymptotic dimension and a property of Nagata

本文证明了每个渐近维数至多为$n$的度量空间$(X, d)$粗等价于满足Nagata性质的度量空间$(Y, D)$：对于$Y$中的每个$n+2$点$y_1,..., y_{n+2}$和对于$Y$中的每个$x$，存在两个不同的$i,j$，使得$D(y_i,y_j)\le D(x,y_i)$ .这解决了《拓扑学II中的开放问题》一书中的第1400个问题。

扩张图阻止性质 (H)

Expanders prevent Property (H)

本文证明扩张图序列的等粗嵌入会破坏巴拿赫空间的性质 (H)，从而排除具有性质 (H) 的空间成为所有可数群的粗泛空间，并给出 Lean 验证。

弯曲锥和性质A

Warped cones and property A

我们描述了一个建设（“翘曲锥建设”），产生的例子粗糙的空间与大集团的翻译。我们表明，通过这种结构，我们可以得到许多不具有性质A或不一致嵌入到希尔伯特空间的粗糙空间的例子。

粗糙顺应性与仿紧性

Coarse amenability versus paracompactness

最近的研究在粗糙的几何揭示了某些概念之间的相似性分析，大规模几何和拓扑结构。G.Yu的性质A是群的顺从性的粗略模拟，它的推广（精确空间）后来被加强为使用单位分割的仿紧空间的大规模模拟。在这篇文章中，我们将更深入地揭示粗糙柔顺性和仿紧性之间的类比。特别是，我们定义了一个新的粗糙模拟仿紧建模的定义特征的扩展。这个类比给出了一个简单的证明，证明了三类空间是粗略不服从的：扩展序列，围长接近无穷大的图空间，以及有限非平凡群的幂的并集。

粗粒度立方体刚性

Coarse cubical rigidity

研究显示，许多右角艺术和科克塞特群的所有紧致立方体划分在粗粒度上看起来相同，诱导相同的粗粒度中位结构。该研究为非超几何群提供了此类性质的第一个例子。

Generic coarse geometry of leaves

A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same constants by some given set of leaves. For instance, the following properties are proved. Either all dense leaves without holonomy are equi-coarsely quasi-isometric to each other, or else there exist residually many dense leaves without holonomy such that each of them is coarsely qu…

Coarse $\mathcal{Z}$-Boundaries for Groups

We generalize Bestvina's notion of a $\mathcal{Z}$-boundary for a group to that of a "coarse $\mathcal{Z}$-boundary." We show that established theorems about $\mathcal{Z}$-boundaries carry over nicely to the more general theory, and that some wished-for properties of $\mathcal{Z}$-boundaries become theorems when applied to coarse $\mathcal{Z}$-boundaries. Most notably, the property of admitting a coarse $\mathcal{Z}$-boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by introducing the notion of a "model $\mathcal{Z}$-geometry." In acco…

Approximate Fixed Point Property for Digital Trees and Products

We add to our knowledge of the approximate fixed point property (AFPP) in digital topology. We show that a digital image that is a tree has the AFPP. Given two digital images (X, κ) and (Y, λ) that have the approximate fixed point property, does their Cartesian product have the AFPP? We explore conditions that yield an affirmative answer. A general answer to this question is not known at the current writing.

某些扩张子和Gromov怪物群的高指数理论II

Higher index theory for certain expanders and Gromov monster groups II

本文是系列文章的第二篇，我们继续研究膨胀子的高阶指数理论。证明了：若图序列的围长趋于无穷大，则极大粗Baum-Connes装配映射是度量空间$X$的同构.正如在本系列的第一篇论文中所讨论的，这可以应用于“格罗莫夫怪物”群的鲍姆-康纳斯猜想。我们还引入了一个新的性质，“几何性质（T”。对于与图序列相关联的度量空间，该性质是最大粗组装映射同构的障碍。这使我们能够区分围长趋于无穷大的扩张子和例如由性质（T）群构造的扩张子。

树上的拟作用与性质（QFA）

Quasi-actions on trees and Property (QFA)

本文证明了树上的拟作用的一些一般性结果，并定义了类似于Serre性质的性质（QFA），但是是在粗糙的条件下。这一属性被证明为一类组，包括$SL(n,\Z)$为$n\geq 3$。我们还提供了一种思考属性（QFA）的方法，将其分解为关于特定树类的语句。

关于有限生成不动子群性质的一个注记

A note on the finitely generated fixed subgroup property

本文借助 BNS 不变量刻画 $G\times\mathbb{Z}^m$ 型群的自同构有限生成不动子群性质，提供部分判定结果与非平凡例子。

Jordan property for homeomorphism groups and almost fixed point property

We study properties of continuous finite group actions on topological manifolds that hold true, for any finite group action, after possibly passing to a subgroup of index bounded above by a constant depending only on the manifold. These include the Jordan property, the almost fixed point property, as well as bounds on the discrete symmetry group. Most of our results apply to manifolds satisfying some restriction such as having nonzero Euler characteristic or having the integral homology of a sphere. For an arbitrary topological manifold $X$ such that $H_*(X;{\mathbf Z})$ is finitely generated,…