Matrix algebra of sets and variants of decomposition complexity

Abstract

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 .

APD profiles and transfinite asymptotic dimension

“…In his paper [3] J. Dydak defined so called APD profile of an ∞-pseudometric space. It is justified and convenient to deal only with integral APD profiles.…”

“…It was proved in [3] that having an APD profile is a hereditary coarse invariant and so is the minimal length of APD profiles. A space X has asymptotic dimension at most n iff (1, n) is an APD profile of X.…”