Extension properties of asymptotic property C and finite decomposition complexity

Beckhardt, Susan; Goldfarb, Boris

doi:10.1016/j.topol.2018.02.034

Cited by 4 publications

(5 citation statements)

References 17 publications

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“…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].…”

Section: Introductionmentioning

confidence: 89%

Decomposition theorems for asymptotic property C and property A

Bell

Głodkowski

Nagórko

2019

Topology and its Applications

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We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APCdecomposition 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 APCdecomposition 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.

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Section: Introductionmentioning

confidence: 89%

Decomposition theorems for asymptotic property C and property A

Bell

Głodkowski

Nagórko

2019

Topology and its Applications

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“…To prove (2), assume that i is given, 1 ≤ i ≤ n. Observe that elements of V i are flat and uniformly bounded directly from the definition. So, it remains only to show that V i is R i -disjoint.…”

Section: 2mentioning

confidence: 99%

“…Several authors have studied the so-called permanence properties -that is, the extent to which properties are preserved by unions, products, etc -of coarse invariants such as finite asymptotic dimension [3], finite decomposition complexity [19,15], or property A [7]. The permanence properties of asymptotic property C have proved to be slightly more elusive than others, with incremental special cases being proved for unions [5] and certain types of group extensions and products [1,2]. Indeed, although it had been conjectured for some time, only recently was a group with infinite asymptotic dimension and asymptotic property C shown to exist [30].…”

Section: Direct Products Preserve Asymptotic Property Cmentioning

confidence: 99%

“…Asymptotic property C is a coarse invariant of metric spaces introduced by A. N. Dranishnikov in 2000 [13]. Recently it received a lot of attention [1,2,5,12,14,15,30].…”

Section: Introductionmentioning

confidence: 99%

“…which (as we show) satisfies assumptions of the Fibering Lemma if K has finite asymptotic dimension and H has asymptotic property C. These observations are summarized in Theorem 1.2. Item (2) in Theorem 1.2 settles the question of whether asymptotic property C is preserved by free products, which was posed in [5]. We do not consider the more general question of free products with an amalgamated subgroup.…”

Section: Introductionmentioning

confidence: 99%

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On the stability of asymptotic property C for products and some group extensions

Bell

Nagórko²

2018

Algebr. Geom. Topol.

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We show that Dranishnikov's asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if G and H are groups with asymptotic property C, then both G×H and G * H have asymptotic property C. We also prove that a group G has asymptotic property C if 1 → K → G → H → 1 is exact, if asdim K < ∞, and if H has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk [15], of Bell and Moran [5], and an open problem in topology in [23].2010 Mathematics Subject Classification. 54F45 (primary), 20F69 (secondary).

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Matrix algebra of sets and variants of decomposition complexity

Dydak

2019

Rev Mat Complut

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We introduce matrix algebra of subsets in metric spaces and we apply it to improve results of Yamauchi and Davila regarding Asymptotic Property C. Here is a representative result: Suppose X is an ∞-pseudo-metric space and n ≥ 0 is an integer. The asymptotic dimension asdim(X) of X is at most n if and only if for any real number r > 0 and any integer m ≥ 1 there is an augmented m× (n + 1)-matrix M = [B|A] (that means B is a column-matrix and A is an m × n-matrix) of subspaces of X of scale-r-dimension 0 such that M ·∩ M T is bigger than or equal to the identity matrix and B(A, r) ·∩ B(A, r) T is a diagonal matrix. IntroductionThis paper is devoted to decomposition complexity understood as any coarse invariant defined in terms of decomposing spaces into r-disjoint subsets. Historically, the first such invariant, namely asymptotic dimension, was introduced by Gromov for the purpose of studying groups using geometric methods (see [32]). In Ostrand ([26] or [27]) formulation (see [4]) it can be defined as follows:Definition 1.1. Suppose X is a metric space. The asymptotic dimension of X is at most n if, for every real number r > 0, there is a decomposition of X into a union of its subsets X 0 , . . . , X n such that each X i is the union of a uniformly bounded and r-disjoint family U i . That means there is a real number S > 0 with each member of U i being of diameter at most S and the distance between points belonging to different elements of U i is at least r.Since then several concepts related to asymptotic dimension were introduced by various authors. One can see them as a spectrum with asymptotic dimension being the strongest concept and weak coarse paracompactness being the weakest (see [7]). The concept closest to asymptotic dimension was introduced by Dranishnikov [9] under the name of Asymptotic Property C: Definition 1.2. Suppose X is a metric space. X has Asymptotic Property C if, for every sequence of real numbers r i > 0, there is a decomposition of X into a finite union of its subsets X 0 , . . . , X n for some natural n such that each X i is the union of a uniformly bounded and r i -disjoint family U i .

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Extension properties of asymptotic property C and finite decomposition complexity

Cited by 4 publications

References 17 publications

Decomposition theorems for asymptotic property C and property A

Decomposition theorems for asymptotic property C and property A

On the stability of asymptotic property C for products and some group extensions

Matrix algebra of sets and variants of decomposition complexity

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