Coarse property C and decomposition complexity

Abstract

Abstract. The coarse category was established by Roe [20] to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov [13]. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C [6] and decomposition complexity [16]. We prove that coarse property C implies coarse property A; we also show that these coarse versions share many of the features of their metric analogs such as preservation by products or unions.

Extension properties of asymptotic property C and finite decomposition complexity

“…There is a recent effort in extending geometric properties of metric spaces to general coarse properties of coarse structures, motivated by applications in controlled K-theory of rings and C * -algebras. This was done for asymptotic dimension by Grave [13] and for APC and FDC by Bell, Moran, and Nagórko [4]. The notion of coarse actions is precisely what is needed to formulate group actions on coarse structures, and we expect generalizations of our results to be true when restated for the coarse properties.…”

Decomposition theorems for asymptotic property C and property A

“…Both APC and FDC have received a lot of attention recently [1,2,4,5,6,10,11,13,18,20]. The current paper is a natural continuation of the decomposition lemma that enabled the first and third authors to prove that asymptotic property C is preserved by free products [6].…”

Coarse free products

“…In this section, we adapt the argument that metric free products preserve asymptotic property C [7] to show that coarse property C [3] is preserved by coarse free products. The main obstacle in proving this is that while in the metric case one has a sequence of numbers R 1 , R 2 , .…”

“…Definition 6.1. [3] A coarse space X has coarse property C if and only if for any sequence E 1 ⊂ E 2 ⊂ · · · of entourages there is a finite sequence U 1 , U 2 , . .…”