arXiv research

Potential theory extended to Gromov hyperbolic spaces.

Boundary rigidity defined for hyperbolic spaces, tied to geometric properties.

Extended Otal-Peigné's Theorem to Gromov-hyperbolic spaces.

Survey solves curvature problems with hyperbolic spaces.

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar resul…

A real valued function $\varphi$ of one variable is called a metric transform if for every metric space $(X,d)$ the composition $d_\varphi = \varphi\circ d$ is also a metric on $X$. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $\varphi$ such that the trans…

Study Gromov hyperbolic domains in Minkowski space, proving equivalence to boundary properties.

Study compares hyperbolic and quasihyperbolic metrics in plane domains.

We prove that the Teichmuller Space of Riemann Surfaces of genus g>1, equipped with the Teichmuller metric, is not a Gromov Hyperbolic space.

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…

Spaces with similar long paths have similar shapes.

This paper extends boundary embedding results to coarsely convex spaces.

Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.

Sharp bounds on hyperbolic metrics in Ptolemaic spaces are derived.

Study examines large deviations in random walks on hyperbolic spaces.

We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.

The study shows that certain curve graphs are hierarchically hyperbolic but not Gromov hyperbolic.

Let S be a surface with genus g and n boundary components and let d(S) = 3g-3+n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmuller space Teich(S) is Gromov-hyperbolic if and only if d(S) …

Proves equivalence of two types of representations of free groups in hyperbolic spaces.

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.

Study shows rigidity of polyhedrons in hyperbolic spaces.

The study constructs AdS manifolds from Gromov-Thurston manifolds.

The study characterizes quasiperiodic surfaces in pseudo-hyperbolic spaces with curvature conditions.

Proves equivalence of two types of boundaries in metric spaces.

Explains visual metrics on hyperbolic space boundaries.

We characterize the class of Gromov hyperbolic spaces, whose boundary at infinity allow canonical Möbius structures.

Study on representations of four-punctured sphere group in hyperbolic spaces.

Study of groups and their quasi-isometrically embedded subgroups.

Space of hyperbolic surfaces is path-connected.

We prove that the embedding of the quaternionic hyperbolic disc $H^1_\mathbb{H}$ into quaternionic hyperbolic $n$-space $H^n_\mathbb{H}$ is tight and thereby obtain the value of the Gromov norm of the quaternionic Kähler class.

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bic…

We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a locally compact, Gromov hyperbolic, ${\rm CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist-Hulin.

We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-mani…

We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these …

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^…

We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. Corollary: the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.

This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Marti…

We introduce the quasi-hyperbolicity constant of a metric space, a rough isometry invariant that measures how a metric space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in the closed interval $[1,2]$. The quasi-hyperbolicity constant of an unbounded Gromov hyperbolic space is equal to…

New method connects CAT(0) spaces to hyperbolic spaces.

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the co…

The study proves Gromov hyperbolicity for certain complex domains.

The paper embeds non-arithmetic hyperbolic manifolds into higher-dimensional spaces.

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk i…

Random walks on hyperbolic spaces show linear growth in translation lengths.

We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to a conjecture of Gabai. Soma has already proven this conjecture earlier. Our tec…

The paper examines hyperbolicity in bounded strongly minimally convex domains in R^d.

Study of Bowditch representations in hyperbolic spaces with implications for dynamics and recognition.