Related papers: Cubulating Infinity in Hierarchically Hyperbolic S…

Let $G$ be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics is compact then $G$ is a hyperbolic group. Let $\mathcal{H}$ be a finite…

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental…

K\"unzi and Yildiz introduced convexity structures in the sense of Takahashi for $T_{0}$-quasi-metric spaces. In this article, we continue this line of study on the Isbell-convex hull of an asymmetrically normed real vector space. Using the…

This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…

We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex,…

We study boundary controllability of one-dimensional coupled hyperbolic-parabolic cascades, focusing on the fine structure of reachable sets. The main model is a wave-heat cascade in which a boundary control acts on the wave equation and…

Pogorelov's rigidity theorem states that a compact convex body in the hyperbolic 3-space is determined up to isometry by the intrinsic path metric on its boundary. The main result of this paper addresses a rigidity problem for non-compact…

We study the metric and topological properties of the space $\mathscr{D}(G)$ of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group $G$ that are quasi-isometric to a word metric, up to rough similarity. This space…

A horoboundary is one of the attempts to compactify metric spaces, and is constructed using continuous functions on metric spaces. It is a concept that includes global information of metric spaces, and its correspondence with an ideal…

A finite-dimensional CAT(0) cube complex $X$ is equipped with several well-studied boundaries. These include the Tits boundary (which depends on the CAT(0) metric), the Roller boundary (which depends only on the combinatorial structure),…

Let $M$ be a complete Riemannian $3$-manifold with sectional curvatures between $0$ and $1$. A minimal $2$-sphere immersed in $M$ has area at least $4\pi$. If an embedded minimal sphere has area $4\pi$, then $M$ is isometric to the unit…

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as…

In complex dynamics, the boundaries of higher dimensional hyperbolic components in holomorphic families of polynomials or rational maps are mysterious objects, whose topological and analytic properties are fundamental problems. In this…

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface…

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as…

Suppose that $X$ is an infinite, connected, locally finite, quasi-transitive graph with the property that every bi-infinite quasi-geodesic uniformly coarsely separates $X$ into exactly two deep pieces. We show that such an $X$ is…

We show that every $n$-quasiflat in a $n$-dimensional $CAT(0)$ cube complex is at finite Hausdorff distance from a finite union of $n$-dimensional orthants. Then we introduce a class of cube complexes, called {\em weakly special} cube…

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are…

We look at isometric actions on arbitrary hyperbolic spaces of generalised Baumslag - Solitar groups of arbitrary dimension (the rank of the free abelian vertex and edge subgroups). It is known that being a hierarchically hyperbolic group…

The Kruskal-Katona theorem together with a theorem of Razborov determine the closure of the set of points defined by the homomorphism density of the edge and the triangle in finite graphs. The boundary of this region is a countable union of…