A. Brudnyi, Yu. Brudnyi
A metric space has the universal Lipschitz extension property if for each subspace S embedded quasi-isometrically into an arbitrary metric space M there exists a continuous linear extension of Banach-valued Lipschitz functions on S to those on all of M. We show that the finite direct sum of Gromov hyperbolic spaces of bounded geometry is universal in the sense of this definition.
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Extension of Gromov's Lipschitz order to with additive errors
本文将 Gromov 的 Lipschitz 序推广到带加性误差的情形，证明其基本性质，并建立其与加性误差意义下 1-Lipschitz 映射的联系。
We mainly consider two metrics: a Gromov hyperbolic metric and a scale invariant Cassinian metric. We compare these two metrics and obtain their relationship with certain well-known hyperbolic-type metrics, leading to several inclusion relations between the associated metric balls.
In this paper we introduce a notion of the Gromov-Hausdorff distance with boundary, denoted by $d_{GHB}$, to construct a framework of convergence of noncomplete metric spaces. We show that a class of bounded $A$-uniform spaces with diameter bounded from below is a complete metric space with respect to $d_{GHB}$. As an application we show the stability of Gromov hyperbolicity, roughly starlike property, uniformization, quasihyperbolization, and boundary of Gromov hyperbolic spaces under appropriate notions of convergence and assumptions.
In this paper, we study certain applications of sphericalization in Gromov hyperbolic metric spaces. We first show that the doubling property regarding two classes of metrics on the Gromov boundary of hyperbolic spaces are coincided. Next, we obtain a characterization of unbounded Gromov hyperbolic domains via metric spaces sphericalization. Finally, we investigate the topological equivalence of Gromov hyperbolic $φ$-uniform domains between the Gromov boundary and the inner metric boundary.
Gromov-Hausdorff convergence of maximal Gromov hyperbolic spaces and their boundaries
研究最大Gromov超几何空间及其边界的连续性性质，证明准度量反向空间序列的近等距收敛等价于最大Gromov超几何空间的Gromov-Hausdorff收敛，且边界等连续性假设下边界也收敛。
Quasihyperbolic metric and Gromov hyperbolic spaces I
本文引入短弧与长度映射概念，利用其性质刻画拟双曲度量空间的Gromov双曲性。
In this note we show that given two complete geodesic Gromov hyperbolic spaces that are roughly isometric and $\varepsilon>0$, either the uniformization of both spaces with parameter $\varepsilon$ results in uniform domains, or else neither uniformized space is a uniform domain. The terminology of "uniformization" is from the work of Bonk, Heinonen and Koskela, where it is shown that the uniformization, with parameter $\varepsilon>0$, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided $\varepsilon$ is small enough.
By a geodesic subspace of a metric space $X$ we mean a subset $A$ of $X$ such that any two points in $A$ can be connected by a geodesic in $A$. It is easy to check that a geodesic metric space $X$ is an $\mathbb{R}$-tree (that is, a $0$-hyperbolic space in the sense of Gromov) if and only if the union of any two intersecting geodesic subspaces is again a geodesic subspace. In this paper, we prove an analogous characterization of general Gromov hyperbolic spaces, where we replace geodesic subspaces by quasigeodesic subspaces.
Gromov hyperbolicity of intrinsic metrics from isoperimetric inequalities
本文利用线性等周不等式刻画Gromov双曲性，证明具有边界扩张性质的Kobayashi、Hilbert及极小度量均为Gromov双曲，并给出新证明及极小度量双曲凸域的刻画。
We prove that a uniformly coarsely proper hyperbolic cone over a bounded metric space consisting of a finite union of uniformly coarsely connected components each containing at least two points is non-amenable and apply this to visual Gromov hyperbolic spaces.
In this paper, we consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. The quasi-invariance properties of both the metrics under quasiconformal maps are also established.
We prove an inequality concerning isometries of a Gromov hyperbolic metric space, which does not require the space to be proper or geodesic. It involves the joint stable length, a hyperbolic version of the joint spectral radius, and shows that sets of isometries behave like sets of $2 \times 2$ real matrices. Among the consequences of the inequality, we obtain the continuity of the joint stable length and an analogue of Berger-Wang theorem.
We characterize Gromov hyperbolicity of the quasihyperbolic metric space (Ω,k) by geometric properties of the Ahlfors regular length metric measure space (Ω,d,μ). The characterizing properties are called the Gehring--Hayman condition and the ball--separation condition.
We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain.
Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance
本文在$\mathbb C^d$中引入局部可见与局部Gromov双曲域概念，证明有界域的局部与全局性质等价，从而可通过边界局部信息检测Gromov双曲性及双全纯映射的连续延拓性。
The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces. While even approximating the distance up to any practical factor poses an NP-hard problem, its relaxations have proven useful for the problems in geometric data analysis, including on point clouds, manifolds, and graphs. We investigate the modified Gromov--Hausdorff distance, a relaxation of the standard distance that retains many of its theoretical properties, which includes their topological equivalence on a rich set of families of metric spaces. We show that the two distances are Lipschitz-equival…
In a recent paper Chatterji and Niblo proved that a geodesic metric space is Gromov hyperbolic if and only if the intersection of any two closed balls has uniformly bounded eccentricity. In their paper, the authors raise the question whether a geodesic metric space with the property that the intersection of any two closed balls has eccentricity 0, is necessarily a real tree. The purpose of this note is to answer this question affirmatively. We also partially improve the main result of Chatterji and Niblo by showing that already sublinear eccentricity implies Gromov hyperbolicity.
Coarse and bi-Lipschitz embeddability of subspaces of the Gromov-Hausdorff space into Hilbert spaces
研究了Gromov-Hausdorff空间子空间在希尔伯特空间中的粗嵌入与双Lipschitz嵌入性，证明了有限点数的子空间可粗嵌入，但无界点数或有限直径的子空间无法双Lipschitz嵌入。
Warped product spaces: Gromov hyperbolicity and identification of the visual boundary
本文针对满足指数增长条件的扭曲积空间，证明其为格罗莫夫双曲空间并推导双曲性常数估计，建立其格罗莫夫边界与 $\partial_GX\times Y$ 的同胚及可视度量比较公式。
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.
Given a complete Gromov hyperbolic space $X$ that is roughly starlike from a point $ω$ in its Gromov boundary $\partial_{G}X$, we use a Busemann function based at $ω$ to construct an incomplete unbounded uniform metric space $X_{\varepsilon}$ whose boundary $\partial X_{\varepsilon}$ can be canonically identified with the Gromov boundary $\partial_ωX$ of $X$ relative to $ω$. This uniformization construction generalizes the procedure used to obtain the Euclidean upper half plane from the hyperbolic plane. Furthermore we show, for an arbitrary metric space $Z$, that there is a hyperbolic filling…
Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces
本文证明了作用在支撑凸测地双组合的堆叠Gromov双曲空间上的离散群的定量Tits'替代，给出了群几何特征的一致估计，并研究了极限下的群作用行为以提供度量空间紧致类的新例子。
Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces
该研究针对Gromov双曲空间边界的函子式莫比乌斯双曲填充问题，对满足对径性质的拟度量对径空间构造了极大边界连续Gromov双曲填充，证明了两类空间的范畴等价及Gromov乘积空间单射性与极大性的等价性。
A real valued function $φ$ of one variable is called a metric transform if for every metric space $(X,d)$ the composition $d_φ= φ\circ d$ is also a metric on $X$. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $φ$ such that the transformed Euclidean half line $([0,\infty),|\cdot|_φ)$ is Gromov hyperbolic. As a consequence, we obtain metric transform rigidity for roughly geodesic Gromov hyperbolic spaces, that is, if $(X,d)$ is any metric space containing a rough geodesic ray and $φ$ is an approximately nondecreasing, unbounded metri…
We give exact and approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We observe that computing the Gromov hyperbolicity from a fixed base-point reduces to a (max,min) matrix product. Hence, using the (max,min) matrix product algorithm by Duan and Pettie, the fixed base-point hyperbolicity can be determined in O(n^2.69) time. It follows that the Gromov hyperbolicity can be computed in O(n^3.69) time, and a 2-approximation can be found in O(n^2.69) time. We also give a (2 log_2 n)-approximation algorithm that runs in O(n^2) time, based on a tree-…
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov b…
On the Gromov hyperbolicity of the minimal metric
本文研究$\u211d^d$中有界强极小凸域关于最小度量的Gromov双曲性，证明其Gromov紧化等价于欧几里得闭包，并探讨了与Hilbert度量的关系。
Uniformization of intrinsic Gromov hyperbolic spaces
本文针对非测地、非紧合的Gromov双曲空间提出一致化方法，证明其共形形变为有界一致空间，且Gromov边界与形变空间度量边界存在自然拟等距，推广了已有经典结果。
Uniformization of Gromov hyperbolic domains by circle domains
本文证明了黎曼球面上的域为Gromov双曲当且仅当共形等价于均匀圆域，解决了Bonk-Heinonen-Koskela猜想并验证了Koebe猜想，且均匀化映射在Möbius变换下唯一。
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.
It is known that the $\tilde j$-metric, half-apollonian metric and scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point) and the supremum is taken over all boundary points. The aim of this paper is to show that taking the average instead of the supremum yields a metric that preserves the Gromov hyperbolicity. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of metrics used in taking the average. We also provide an …
In [BBM21], Belk, Bleak and Matucci proved that hyperbolic groups can be seen as subgroups of the rational group. In order to do so, they associated a tree of atoms to each hyperbolic group. Not so many connections between this tree and the literature on hyperbolic groups were known. In this paper, we prove an atom-version of the fellow traveler property and exponential divergence, together with other similar results. These leads to several consequences: a bound from above of the topological dimension of the Gromov boundary, the definition of an augmented tree which is quasi-isometric to the C…
Polyhedral structure of maximal Gromov hyperbolic spaces with finite boundary
该研究刻画了有限边界的极大Gromov双曲空间的多面体结构，证明其等距于嵌入ℝⁿ的有限多面体复形，还给出了这类空间为单射度量空间的简化证明，并定义了相关的Teichmuller空间。
For a non-elementary discrete isometry group $G$ of divergence type acting on a proper geodesic $δ$-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of $G$. As applications of this result, we have: (1) under a minor assumption, such a discrete group $G$ admits no proper conjugation, that is, if the conjugate of $G$ is contained in $G$, then it coincides with $G$; (2) the critical exponent of any non-elementary normal subgroup of $G$ is strictly greater than half of that for $G$.
In this paper we study asymptotic properties of symmetric and non-degenerate random walks on transient hyperbolic groups. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous results by S. Sawyer and T. Steger and F. Ledrappier for certain CAT minus one groups. The proofs use a result by A. Ancona on the identification of the Martin boundary of a hyperbolic group with its Gromov boundary. We also give a new interpretation, in terms of Hilbert metrics, of the Green metric, first introduced by S. Brofferio and S. Blachere.
We generalize the recent results of Björn-Björn-Shanmugalingam \cite{BBS20} concerning how measures transform under the uniformization procedure of Bonk-Heinonen-Koskela for Gromov hyperbolic spaces \cite{BHK} by showing that these results also hold in the setting of uniformizing Gromov hyperbolic spaces by Busemann functions that we introduced in \cite{Bu20}. In particular uniformly local doubling and uniformly local Poincaré inequalities for the starting measure transform into global doubling and global Poincaré inequalities for the uniformized measure. We then show in the setting of uniform…
We use bounded cohomology to define a notion of volume of an SO(n,1)-valued representation of a lattice SO(n,1) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff in this setting. Our approach gives in particular a proof of Thurston's version of Gromov's proof of Mostow Rigidity (also in the non-cocompact case), which is dual to the Gromov--Thurston proof using the simplicial volume invariant.}
If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-hyperbolic (in the Gromov sense) if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In the context of graphs, to remove and to contract an edge of a gra…
Discrete-time gradient flows in Gromov hyperbolic spaces
本文研究了Gromov双曲空间上Lipschitz凸函数的近端点算法，证明了其能从任意初始点逼近最小化子，并建立了近端算子的收缩估计，适用于树的小扰动。
Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a "worst-case" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation from tree-like structure. In this article we introduce an "average-case" version of Gromov hyperbolicity, which detects whether the "most of the space", with respect to a given probability measure, looks like a tree. The main result of the paper is that if this average hyperbolicity is small, then the space can be approximately embedded in a tree. The proof u…
In this paper we prove that random $d$--regular graphs with $d\geq 3$ have traffic congestion of the order $O(n\log_{d-1}^{3}(n))$ where $n$ is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically $δ$--hyperbolic for any non--negative $δ$ almost surely as $n\to\infty$.
Rough isometry between Gromov hyperbolic spaces and unbounded uniformization
本文证明了两个proper、roughly starlike的Gromov双曲空间在粗等距下，其共形变形的一致性是同时性质，为Shanmugalingam和Lindquist的工作提供了对应。
Quasi-symmetries between metric spaces and rough quasi-isometries between their infinite hyperbolic cones
本文研究度量空间拟对称与无限双曲锥粗糙拟等距之间的对应关系，证明了三类推广结果：幂拟对称诱导粗糙拟等距、完备度量空间可实现为无限双曲锥的相对 Gromov 边界、可视 Gromov 双曲空间与其边界的无限双曲锥粗糙相似，从而将文献 [3] 中相关定理推广到无界空间和无限锥情形。
We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous l_p-cohomology, thereby obtaining information about the l_p-equivalence relation, as well as critical exponents associated with l_p-cohomology. As an application, we provide a flexible construction of hyperbolic groups which do not have the Combinatorial Loewner Property, extending and complementing earlier examples. Another consequenc…
Barycenters and a law of large numbers in Gromov hyperbolic spaces
本文研究Gromov双曲空间中概率测度的重心，证明收缩性质、有限均匀分布重心的确定性逼近和大数定律，将CAT(0)空间结果推广至含双曲常数附加项的情形。
Characterizing Hyperbolicity in Graphs
研究有限图双曲性刻画问题，引入与格罗莫夫δ及直径相关函数定义归一化不变量，推导出双曲平面上该函数的闭式公式，据此给出双曲平面格罗莫夫δ-双曲性最佳常数的新证明及直径缩放下标度化格罗莫夫四点条件最优常数的首个理论证明。
For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor $K$. This setup was taken up to study the probabilistic potential theory on $K$, and the bi-Lipschitz equivalence on $K$. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and …
Gromov hyperbolic domains in Minkowski space
研究闵可夫斯基空间中格罗莫夫超几何域的性质，通过马尔科维茨度量与因果边界的关系，揭示其与稳定无因果性的等价性。
If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $δ(X)$ the sharp hyperbolicity constant of $X$, i.e. $δ(X)=\inf\{δ\ge 0: \, X \, \text{ is $δ$-hyperbolic}\}.$ In this paper we characterize the lexicographic product of two graphs $G…
Otal-Peigné's Theorem for Gromov-hyperbolic spaces
将Otal-Peigné定理推广至线凸的真Gromov-双曲空间，证明离散几乎自由等距作用群的临界指数等于商空间测地流拓扑熵，并给出反例说明线凸性条件不可去除。
We consider a `contracting boundary' of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space. We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.
Lipschitz distance between clouds
本文证明了单点度量空间和实数线到有限Gromov-Hausdorff距离的度量空间类的Lipschitz距离为正，揭示了这些类之间的分离性。
Gromov hyperbolicity III: an improved geometric characterization and its applications
本文通过新的方法建立了Gromov超几何性的几何刻画，并在欧几里得空间中解决了Balogh-Buckley问题，同时改进了先前结果并拓展了应用。
Uniformizing non-proper Gromov Hyperbolic Spaces
本研究将经典单值化理论推广至非紧非测地长度空间，证明完备粗星型Gromov双曲空间拟等距类与有界一致空间拟相似类一一对应，解决了公开问题，核心创新是用$(c,μ)$-拟测地线替代拟双曲测地线。
Discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces
本文研究 Gromov 双曲空间中无界凸函数的离散时间梯度流，证明其沿唯一负渐近斜率方向发散，并给出收敛速率估计。
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 class of visual functions $f$ that do induce continuous maps $\tilde f$ between Gromov boundaries. Its subclass, the class of radial functions, induces Hoelder maps between Gromov boundaries. Conversely, every Hoelder map between Gromov boundaries of visual hyperbolic spaces induce…
The uniformization and hyperbolization transformations formulated by Bonk, Heinonen and Koskela in \emph{"Uniformizing Gromov Hyperbolic Spaces"}, Astérisque {\bf 270} (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local $p$-Poincaré inequality, then the transformed mea…
Gromov hyperbolization of unbounded noncomplete spaces and Hamenstädt metric
本文针对无界、局部完备但不完备的度量空间，考察其在三类双曲型度量下的 Gromov 双曲化，并据此给出 Banach 空间中无界一致域的 Gromov 双曲刻画。
We prove that any measured Gromov-Hausdorff precompact set of metric measure spaces which is contained in a certain set, called a pyramid, is bounded by some metric measure space with respect to the Lipschitz order inside the pyramid. This is proved as a step towards a possible extension of the statement of Gromov, for which we gave a detailed proof in our previous work. Several related results are also obtained.
In this paper, we prove an exponential and Ganssian concentration inequality for 1-Lipschitz maps from mm-spaces to Hadamard manifolds. In particular, we give a complete answer to a question by M. Gromov.
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