Related papers: Rigidity of Higson coronas

This paper deepens into the relations between coarse spaces and compactifications, by defining a $C_0$ coarse structure attached to a family of pseudometrics. This definition allow us to give a more topological point of view on the…

To a metric space $X$ we associate a compact topological space $\nu' X$ called the corona of $X$. Then a coarse map $f:X\to Y$ between metric spaces is mapped to a continuous map $\nu' f:\nu' X\to \nu' Y$ between coronas. Sheaf cohomology…

This paper is devoted to introducing coarse structures in a very simple way, namely as an equivalence relation on the set of simple ends. As an application we show that Gromov boundary of every hyperbolic space is an example of a Higson…

We prove that two homogeneous ultra-metric spaces $X,Y$ are coarsely equivalent if and only if $\mathrm{Ent}^\sharp(X)=\mathrm{Ent}^\sharp(Y)$ where $\mathrm{Ent}^\sharp(X)$ is the so-called sharp entropy of $X$. This classification implies…

This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the…

We investigate the fixed point property of the group actions on a coarse space and its Higson corona. We deduce the coarse version of Brouwer's fixed point theorem.

For a topological space $X$ a topological contraction on $X$ is a closed mapping $f:X\to X$ such that for every open cover of $X$ there is a positive integer $n$ such that the image of the space $X$ via the $n$th iteration of $f$ is a…

Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willett that if the uniform Roe algebras of…

We consider a perturbation $f$ of a hyperbolic toral automorphism $L$. We study rigidity related to exceptional properties of the strong and weak stable foliations for $f$. If the strong foliation is mapped to the linear one by the…

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's…

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse…

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e…

A coarse compactification of a proper metric space $X$ is any compactification of $X$ that is dominated by its Higson compactification. In this paper we describe the maximal coarse compactification of $X$ whose corona is of dimension $0$.…

In this paper, we study the rigidity of uniform Roe algebras via the ideal structures. We showed that for given metric spaces X and Y with bounded geometry, if their uniform Roe algebras are isomorphic, then they are coarse equivalent.

We consider rigidity properties of compact symmetric spaces $X$ with metric $g_0$ of rank one. Suppose $g$ is another Riemannian metric on $X$ with sectional curvature $\kappa$ bounded by $0 \leq \kappa \leq 1$. If $g$ equals $g_0$ outside…

Let $f$ and $g$ be two circle endomorphisms of degree $d\geq 2$ such that each has bounded geometry, preserves the Lebesgue measure, and fixes $1$. Let $h$ fixing $1$ be the topological conjugacy from $f$ to $g$. That is, $h\circ f=g\circ…

We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse…

We establish rigidity results for holomorphic mappings and plurisubharmonic functions in complex geometry. First, under mild conditions, we show that the gradient of a $\operatorname{U}(1)$-invariant strictly plurisubharmonic function in…

Let (M,g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its…

Roe algebras are C*-algebras built using large-scale (or 'coarse') aspects of a metric space (X,d). In the special case that X=G is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (G,d) is…