arXiv research

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

Potential theory extended to Gromov hyperbolic spaces.

The study proves Gromov hyperbolicity for certain complex domains.

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

We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi di…

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

We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry …

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…

In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $C^\infty$ boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also …

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

Compact manifolds with specific cover properties are hyperbolic.

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

Survey solves curvature problems with hyperbolic spaces.

In this paper we prove: if the complete Kähler-Einstein metric on a bounded convex domain (with no boundary regularity assumptions) is Gromov hyperbolic, then the $\bar{\partial}$-Neumann problem satisfies a subelliptic estimate. This is accomplished by constructing bounded plurisubharmonic function whose Hessian grows…

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…

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

We prove, in the context of Hilbert geometry, the equivalence between the existence of an upper bound on the area of ideal triangles and the Gromov-hyperbolicity.

We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.

Spaces with similar long paths have similar shapes.

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…

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.

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.

This paper extends boundary embedding results to coarsely convex spaces.

Study examines large deviations in random walks on hyperbolic spaces.

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.

Characterizes visibility and geodesic loops in complex domains.

Characterizes when geodesics in groups are generic.

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…

Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.

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

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

New examples of 5D manifolds without certain structures.

We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplici…

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

Paper proves vanishing homology groups for certain hyperbolic groups.

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…

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

Classifies mapping tori of specific groups, generalizing known results.

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) …

Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we giv…

This paper shows that every Gromov hyperbolic group can be described by a finite subdivision rule acting on the 3-sphere. This gives a boundary-like sequence of increasingly refined finite cell complexes which carry all quasi-isometry information about the group. This extends a result from Cannon and Swenson in 1998 th…

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

New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.

Study compares hyperbolic and quasihyperbolic metrics in plane domains.

The grand arc graph's asymptotic dimension is shown to be infinite.

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.

Sharp bounds on hyperbolic metrics in Ptolemaic spaces are derived.

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