Coarse property {C} and decomposition complexity

Preserving Coarse Properties

The aim of this paper is to investigate properties preserved and co-preserved by coarsely $n$-to-1 functions, in particular by the quotient maps $X\to X/\sim$ induced by a finite group $G$ acting by isometries on a metric space $X$. The coarse properties we are mainly interested in are related to asymptotic dimension and its generalizations: having finite asymptotic dimension, asymptotic Property C, straight finite decomposition complexity, countable asymptotic dimension, and metric sparsification property. We provide an alternative description of asymptotic Property C and we prove that the cl…

Decomposition theorems for asymptotic property C and property A

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.

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…

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。

Extension properties of asymptotic property C and finite decomposition complexity

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…

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…

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。

Coarse amenability versus paracompactness

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using partitions of unity. In this paper we go deeper into divulging analogies between coarse amenability and paracompactness. In particular, we define a new coarse analog of paracompactness modelled on the defining characteristics of expanders. That analog gives an easy proof of three cate…