Related papers: Semigroup-controlled asymptotic dimension

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces…

We prove a Hurewicz-type theorem for the dynamic asymptotic dimension originally introduced by Guentner, Willett, and Yu. Calculations of (or simply upper bounds on) this dimension are known to have implications related to cohomology of…

We introduce a geometric property complementary-finite asymptotic dimension (coas- dim). Similar with asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem.

We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for…

In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner…

We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups $f:(\Xi, \Xi^\infty) \to (\Lambda, \Lambda^\infty)$,…

We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with…

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the…

In repeated Measure Designs with multiple groups, the primary purpose is to compare different groups in various aspects. For several reasons, the number of measurements and therefore the dimension of the observation vectors can depend on…

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…

In their theorem from 2006, A. Dranishnikov and J. Smith prove that if $f:G\to H$ is a group homomorphism, then the following formula for asymptotic dimension is true: $\operatorname{asdim} G \leq \operatorname{asdim} H +…

In this work we study two problems about Assouad-Nagata dimension: 1) Is there a metric space of non zero Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes) 2) Suppose $G$…

In this paper we present a complete description of a stochastic semigroup of finite-dimensional projections in Hilbert space. The geometry of such semigroups is characterized by the asymptotic behavior of the widths of compact subsets with…

We construct a countable p-local group with a proper invariant metric whose Assouad-Nagata dimension is strictly greater than the asymptotic dimension with linear control. This solves Problem 8.6 from the list [Dr]. We study asymptotic…

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{H_1, ..., H_m\} $. We prove that if each of the subgroups $H_1, ..., H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$…

The asymptotic dimension theory was founded by Gromov in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and…

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,…

Asymptotic methods for hypothesis testing in high-dimensional data usually require the dimension of the observations to increase to infinity, often with an additional condition on its rate of increase compared to the sample size. On the…

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric…