Related papers: Transfinite Asymptotic Dimension
In this paper, we define asymptotic dimension of fuzzy metric spaces in the sense of George and Veeramini. We prove that asymptotic dimension is an invariant in the coarse category of fuzzy metric spaces. We also show several consequences…
We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\'asz Local Lemma. For metric measure spaces with volume doubling…
The notion of asymptotic space for an unbounded metric space has been introduced by Micha Gromov in 1980s. It is intended to capture the structure of a metric space at infinity. The most comprehensive definition of asymptotic space is given…
Given a metric space $X$ of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal $n$ if there is a linear…
We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both omega+1, where omega is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not…
The notion of the decomposition complexity was introduced in \cite{GTY} using a game theoretical approach. We introduce a notion of straight decomposition complexity and compare it with the original as well with the asymptotic property C.…
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…
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however,…
We prove the existence of an upper bound on the asymptotic dimension of tree amalgamations of locally finite quasi-transitive connected graphs. This generalises a result of Dranishnikov for free products with amalgamation and a result of…
Finite decomposition complexity and asymptotic dimension growth are two generalizations of M. Gromov's asymptotic dimension which can be used to prove property A for large classes of finitely generated groups of infinite asymptotic…
We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at…
We define the direct sum of a countable family of pointed metric spaces in a way resembling the direct sum of groups. Then we prove that if a family consists of $0$-hyperbolic (in the sense of Gromov) and $D$-discrete spaces, then its…
We show for a free action of a countable group $\Gamma$ on a finite-dimensional, compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of $\Gamma$.
In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or…
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric…
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…
We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.
It is proven that if a finitely presented group is one ended it has asymptotic dimension bigger than one. It follows that finitely presented groups with asdim 1 are virtually free. A counterexample is given for the finitely generated case.
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 $\infty$-pseudo-metric space and $n\ge 0$…
In this paper we prove that the asymptotic dimension of a finite-dimensional CAT(0) cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT(0) cube…