Related papers: Asymptotic dimension and the disk graph I

This paper investigates quasi-isometries between graphs with variable edge lengths. A quasi-isometry is a mapping between metric spaces that approximately preserves distances, allowing for a bounded amount of additive and multiplicative…

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional…

Statistical analysis of a graph often starts with embedding, the process of representing its nodes as points in space. How to choose the embedding dimension is a nuanced decision in practice, but in theory a notion of true dimension is…

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…

We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative trisections for given 4-manifolds. We prove…

Let S be the boundary of a handlebody M. We prove that the set of curves in S that are boundaries of disks in M, considered as a subset of the complex of curves of S, is quasi-convex.

A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…

An isometric embedding of a graph into a metric space is an embedding of the vertices such that the smallest number of edges connecting any two vertices equals to the distance in the metric space between the images. In this paper, we study…

We show that intersection graphs of compact convex sets in R^n of bounded aspect ratio have asymptotic dimension at most 2n+1. More generally, we show this is the case for intersection graphs of systems of subsets of any metric space of…

We show that if a graph $G$ admits a quasi-isometry $\phi$ to a graph $H$ of bounded path-width, then we can assign a non-negative integer length to each edge of $H$, such that the same function $\phi$ is a quasi-isometry to this weighted…

We provide conditions under which a Riemann surface $X$ is the asymptotic boundary of a convex co-compact hyperbolic manifold, homeomorphic to a handlebody, of negative renormalized volume. We prove that this is the case when there are on…

We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We…

Let $p$ be a real number greater number greater than one. Suppose that a graph $G$ of bounded degree is quasi-isometric with a Riemannian manifold $M$ with certain properties. Under these conditions we will show that the $p$-harmonic…

In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this…

In our previous works, we constructed diffeomorphisms of compact 4-manifolds $X$ by surgeries on theta-graphs embedded in $X$. In this paper, we consider the case $X=M\times I$, where $M$ is a spherical 3-manifold. For some of such $X$, we…

We determine the local geometric structure of two-dimensional metric spaces with curvature bounded above as the union of finitely many properly embedded/branched immersed Lipschitz disks. As a result, we obtain a graph structure of the…

It is well-known that a paracompact space $X$ is of covering dimension at most $n$ if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton $K^{(n)}$. We use the same idea to…

The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical…

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold $M$ to a closed, oriented 3-manifold $N$ if and only if $M$…

Topological surgery in dimension $3$ is intrinsically connected with the classification of $3$-manifolds and with patterns of natural phenomena. In this expository paper, we present two different approaches for understanding and visualizing…