Related papers: Preserving Coarse Properties
We investigate the dynamic asymptotic dimension for \'etale groupoids introduced by Guentner, Willett and Yu. In particular, we establish several permanence properties, including estimates for products and unions of groupoids. We also…
For a discrete metric space (or more generally a large scale space) $X$ and an action of a group $G$ on $X$ by coarse equivalences, we define a type of coarse quotient space $X_G$, which agrees up to coarse equivalence with the orbit space…
We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group $G$ by a family of groups $\mathcal{F}=\{(G_{\alpha},…
This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of…
The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.]…
In this article we survey the development of generic and coarse computability and the main results on how classical asymptotic density interacts with the theory of computability.
Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was…
Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse…
The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. Our recent paper in SoCG 2021 defined isometry invariants (density functions), which are complete in general position and continuous under…
In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to…
Given a non-decreasing function $f \colon \mathbb{N} \to \mathbb{N}$ we define a single player game on (infinite) connected graphs that we call fire retaining. If a graph $G$ admits a winning strategy for any initial configuration (initial…
This paper studies the asymptotic product of two metric spaces. It is well defined if one of the spaces is visual or if both spaces are geodesic. In this case the asymptotic product is the pullback of a limit diagram in the coarse category.…
We formalize the concept of a family of metric spaces satisfying a coarse property uniformly and we generalize finite decomposition complexity of Erik Guentner, Romain Tessera, and Guoliang Yu. Of particular interest are results determining…
In this paper we provide a unified treatment of some convex minimization problems, which allows for a better understanding and, in some cases, improvement of results in this direction proved recently in spaces of curvature bounded above.…
We prove that uniformly locally finite quasigeodesic coarse median spaces of finite rank and at most exponential growth have Property A. This offers an alternative proof of the fact that mapping class groups have property A.
Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one…
In this paper, we study some features of n-normed spaces with respect to norms of its quotient spaces. We define continuous functions with respect to the norms of its quotient spaces and show that all types of continuity are equivalent. We…
The main goal of this note is to prove a coarse analogue of Factorization Theorems in Dimension Theory: Let $f: X \rightarrow Y$ be a coarsely continuous map. Then $f$ factors through coarsely continuous maps $g : X \rightarrow Z$ and $h :…
The primary goal of this document is to record the asymptotic effects that preimage constraints impose upon the sizes of the iterated images of a random function. Specifically, given a subset $\mathcal{P}\subseteq \mathbb{Z}_{\geq 0}$ and a…
Given a (not necessarily discrete) proper metric space $M$ with bounded geometry, we define a groupoid $G(M)$. We show that the coarse Baum--Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M)…