Related papers: Visual maps between coarsely convex spaces

The classifying space of inertial reference frames in special relativity is naturally hyperbolic. There is a remarkable interplay between central elements of hyperbolic geometry and those of special relativity -- which, to a certain extent,…

Let M be a compact hyperbolic manifold with totally geodesic boundary. If the injectivity radius of the boundary is larger than an explicit function of the normal injectivity radius of the boundary, we show that there is a negatively curved…

In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of…

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient…

For a proper geodesic metric space $X$, the Morse boundary $\partial_*X$ focuses on the hyperbolic-like directions in the space $X$. It is a quasi-isometry invariant. That is, a quasi-isometry between two hyperbolic spaces induces a…

This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustive database of maps of a metric space into…

This is an expository survey with two goals. 1) The primary goal is to discuss and highlight the impact of two recent influential ideas in geometric group theory. The first of which is the notion of an injective metric space which is a rich…

This paper develops methods based on coarse geometry for the comparison of wavelet coorbit spaces defined by different dilation groups, with emphasis on establishing a unified approach to both irreducible and reducible quasi-regular…

This is a tale describing the large scale geometry of Euclidean plane domains with their hyperbolic or quasihyperbolic distances. We prove that in any hyperbolic plane domain, hyperbolic and quasihyperbolic quasi-geodesics are the same…

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called…

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean…

We introduce a class of mappings called vertical quasi-isometries and show that branched quasisymmetries $X\to Y$ of Guo and Williams between compact, bounded turning metric doubling spaces admit natural vertically quasi-isometric…

We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient topological space of the metric boundary, and that therefore a word-hyperbolic…

Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of…

We introduce a class of metric spaces which we call "bolic". They include hyperbolic spaces, simply conneccted complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for…

We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we…

We introduce two notions of coarse embeddability between operator spaces: almost complete coarse embeddability of bounded subsets and spherically-complete coarse embeddability. We provide examples showing that these notions are strictly…

In a recent paper, Zhou, Ponnusamy, and Rasila [Math. Nachr. (2025)] have established that the conformal deformations, with parameter $\epsilon>0$, of a Gromov hyperbolic space via Busemann functions are uniform spaces for sufficiently…

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary…

We study Bowditch's notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly…