Related papers: Quasi-isometries between visual hyperbolic spaces

A holomorphic curve in moduli spaces is the image of a non-constant holomorphic map from a hyperbolic surface $B$ of type $(g,n)$ to the moduli space $\mathcal{M}_h$ of closed Riemann surfaces of genus $h$. We show that, when all peripheral…

It is proved the following theorem, if $w$ is a quasiconformal harmonic mappings between two Riemann surfaces with smooth boundary and aproximate analytic metric, then $w$ is a quasi-isometry with respect to Euclidean metric.

We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space. This approach is applicable even if the space does not support any Poincar\'e inequalities. As an application of the…

We give an example of a visual Gromov Hyperbolic metric space X with asdimX=2 and dim(bdry X)=0

The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.

We identify a condition that prevents a hyperbolic space from being quasi-isometric to the curve complex of any non-sporadic surface. Our result applies to several hyperbolic complexes, including arc complexes, disk complexes,…

This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.

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…

This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous…

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups…

The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic-like behavior in the space. A key property of this boundary is that a quasi-isometry between two such spaces induces a homeomorphism on their Morse…

We introduce a new quasi-isometry invariant $\subcorank X$ of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for…

We classify all negatively curved $\R^n \rtimes \R$ up to quasiisometry. We show that all quasiisometries between such manifolds (except when they are biLipschitz to the real hyperbolic spaces) are almost similarities. We prove these…

We show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a cone arc. This result provides a new approach to the elementary metric geometry question, formulated in…

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 give a new characterization of (homogeneous) Triebel-Lizorkin spaces $\dot{\mathcal F}^{s}_{p,q}(Z)$ in the smoothness range $0 < s < 1$ for a fairly general class of metric measure spaces $Z$. The characterization uses Gromov hyperbolic…

We consider the quasihyperbolic metric, and its generalizations in both the $n$-dimensional Euclidean space $R^n$, and in Banach spaces. Historical background, applications, and our recent work on convexity properties of these metrics are…

Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via…

We prove that, if a group is relatively hyperbolic, the parabolic subgroups are virtually nilpotent if and only if there exists a hyperbolic space with bounded geometry on which it acts geometrically finitely. This provides, by use of M.…

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric…