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

Bell, G.; Nagórko, Andrzej

doi:10.2140/agt.2018.18.221

Cited by 6 publications

(19 citation statements)

References 23 publications

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“…In Section 5 we provide an upper bound for the dimension of a coarse free product. We conclude by showing how the arguments of Bell and Nagórko [7] can be adapted to show that coarse property C is preserved by coarse free products in Section 6.…”

Section: Introductionmentioning

confidence: 88%

“…It is not known whether coarse property C satisfies fibering permanence, so we cannot use Theorem A to show that coarse free products preserve coarse property C. We prove this using techniques similar to Bell and Nagórko [7] in Section 6.…”

Section: Combining This With Theorem a Immediately Impliesmentioning

confidence: 99%

“…For ease of notation, we defined the unary free product, following Bell and Nagórko [7]. One can obtain the free product X * Y of pointed spaces (X, x 0 ) and (Y, y 0 ) by forming * (X ⊔ Y /x 0 ∼ y 0 ).…”

Section: The Coarse Free Productmentioning

confidence: 99%

“…Other coarse properties of groups, including finite decomposition complexity [12], property A [5], and asymptotic property C [7], are preserved by the operation of free product. In the cases of finite decomposition complexity and property A, the method of proof has become standard: one applies the Bass-Serre theory, union permanence, and fibering permanence, (see [11]).…”

Section: Introductionmentioning

confidence: 99%

“…In the cases of finite decomposition complexity and property A, the method of proof has become standard: one applies the Bass-Serre theory, union permanence, and fibering permanence, (see [11]). Other techniques were required in the case of asymptotic property C [7].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Coarse free products

Bell

Lawson

2020

Topology and its Applications

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Section: Introductionmentioning

confidence: 88%

Section: Combining This With Theorem a Immediately Impliesmentioning

confidence: 99%

Section: The Coarse Free Productmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coarse free products

Bell

Lawson

2020

Topology and its Applications

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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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Classification of metric spaces with infinite asymptotic dimension

Zhu

2018

Topology and its Applications

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We introduce a geometric property complementary-finite asymptotic dimension (coasdim). Similar with asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem. Moreover, we show that coasdim(X) ≤ f in + k implies trasdim(X) ≤ ω + k − 1 and transfinite asymptotic dimension of the shift union shIn Section 5, we define the shift union ofiZ is no more than ω + 1. Finally, we give a negative answer to the Question 7.1 raised in [17]. PreliminariesOur terminology concerning the asymptotic dimension follows from [3] and for undefined terminology we refer to [11]. Let (X, d) be a metric space and U, V ⊆ X, let diam U = sup{d(x, y) : x, y ∈ U } and d(U, V ) = inf{d(x, y) : x ∈ U, y ∈ V }. Let R > 0 and U be a family of subsets of X, U is said to be R-bounded if diam U △ = sup{diam U : U ∈ U} ≤ R.U is said to be uniformly bounded if there exists R > 0 such that U is R-bounded. Let r > 0, U is said to be r-disjoint if d(U, V ) ≥ r for every U, V ∈ U and U = V.

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

Cited by 6 publications

References 23 publications

Coarse free products

Coarse free products

Matrix algebra of sets and variants of decomposition complexity

Classification of metric spaces with infinite asymptotic dimension

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