Indira Chatterji, Graham A. Niblo
We show that a geodesic metric space is hyperbolic in the sense of Gromov if and only if intersections of balls have bounded eccentricity. In particular, $\R$-trees are characterized among geodesic metric spaces by the property that the intersection of any two balls is always a ball. Both Gromov hyperbolicity and CAT($κ$) geometry can be characterised in terms of the geometry of the intersection of balls.
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We show that Gromov's monster groups arising from i.i.d. labelings of expander graphs do not admit non-elementary actions on geodesic hyperbolic spaces. The proof relies on comparing properties of random walks on randomly labeled graphs and on groups acting non-elementarily on hyperbolic spaces.
Two Characterizations of Geometrically Infinite Actions on Gromov Hyperbolic Spaces
本文通过逃逸测地线和不可数非锥形极限点的存在性，为几何无限作用提供了两种新刻画，扩展了Bonahon、Bishop及Kapovich-Liu在Kleinian群和负曲率流形中的定理。
A note on almost negative matrices and Gromov-hyperbolic Coxeter groups
本文重新审视Moussong关于Gromov双曲Coxeter群的刻画，指出原始论证存在一个小漏洞，并给出补救方法。
We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result generalizes from a unified framework previous works of Arzhantseva-Lysenok and Sambusetti, and provides an answer to a question of the latter.
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 show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are un…
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.
This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for …
We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups).
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 results for the linear filling radius inequality. Our results strengthen and generalize theorems of Gromov, Papasoglu and others.
Coning off a collection of uniformly quasiconvex subsets of a Gromov hyperbolic space leaves a new space, called the cone-off. Kapovich and Rafi generalized work of Bowditch to show this space is still Gromov hyperbolic. We show that the Gromov boundary of cone-off embeds in the boundary of the original hyperbolic space. (A stronger version of this result was previously obtained by Dowdall and Taylor; see Note in text.) Moreover, under some acylindricity assumptions we give a precise description of the image. As an application, we are able to characterize the elliptic and loxodromic elements o…
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.
Coboundary expansion and Gromov hyperbolicity
本文证明了若紧致 $n$-流形允许一列在 $n-2$ 维形成余边界展开的剩余覆盖，则其具有 Gromov 双曲基本群，并指出非双曲紧致连通不可约 3-流形的剩余覆盖列不是 1-余边界展开子。
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 that hyperbolic groups can be described by a recursive sequence of overlapping coverings by possibly wild sets, and demonstrates the existence of non-cubulated groups that can be represented by subdivision rules.
We give a cohomological characterization of Gromov relative hyperbolicity. As an application we prove a converse to the combination theorem for graphs of relatively hyperbolic groups given in a previous paper of the first author. We build upon, and follow the ideas of, the work of S. Gersten in ``Cohom. lower bounds for isoperim. funct. on groups'' (Topology 37, 1998) about the same topics in the classical Gromov hyperbolic setting.
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
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.
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…
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.
Groups admitting Wirtinger presentations and Gromov hyperbolic groups
研究承认扭曲维尔廷格表示的群，证明若此类有限生成群是格罗莫夫双曲的，其第二有理同调群消失，无挠时第二整同调群也消失。
We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskiĭ. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$.
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.}
Trees of Graphs as Boundaries of Hyperbolic Groups
本文刻画了Gromov边界同胚于图的树的1端字双曲群，并通过Bowditch JSJ分裂的代数性质给出了此类具有最简单连通边界的群的等价条件。
We present a careful approximation of the geodesics in trees of hyperbolic or relatively hyperbolic groups. As an application we prove a combination theorem for finite graphs of relatively hyperbolic groups, with both Farb's and Gromov's definitions.
In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the hyperbolicity constant and of an upper bound of the entropy of the space and of an upper bound of the diameter of its quotient. As a consequence we show that the set of non cyclic torsion-free $δ$-hyperbolic marked groups whose entropy is bounded above by a number $H$ is finite with cardinality depending on $δ$ and $H$ alone. From these results, we draw homo…
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…
Marked length spectra of Gromov hyperbolic space
本文证明了在群$Γ$的可比等距作用下，若两个粗测地完备的Gromov双曲空间具有相同的渐近标记长度谱且极限集相同，则它们是$Γ$-等变粗等距的。
In this article we produce an example of a non-residually finite group which admits a uniformly proper action on a Gromov hyperbolic space.
We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic loca…
We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as $G=G_1\ast\dots\ast G_k\ast F$, where $F$ denotes a finitely generated free group, we identify the Gromov boundary of the graph of relative cyclic splittings with the space of equivalence classes of $\mathcal{Z}$-averse trees in the boundary of the corresponding outer space. A tree is \emph{$\mathcal{Z}$-averse} if it is not compatible with any tree $T'$, that is itself comp…
Let $Γ$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $μ$ a probability measure on $Γ$ whose support generates a non-elementary subsemigroup. Under the assumption that $μ$ has a finite exponential moment, we establish large deviations results for the distance and the translation length of a random walk with driving measure $μ$. From our results, we deduce a special case of a conjecture regarding large deviations of spectral radii of random matrix products.
Convex cocompact right-angled Gromov-Thurston polyhedra
针对5维凸余紧双曲反射群的相关问题，本文采用与Lee和Marquis不同的方法，构造了直角情形的对应例子。
These notes are the English version of the paper "Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of Gamma on this graph to find an explicit non trivial quasimorphism on Gamma and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a …
We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a 'negatively curved' group. Another result says that the image of a group, which is boundedly generated by a finite set of amenable subgroups, in a Gromov hyperbolic group has to be virtually cyclic. Moreover, we show that any homomorphic image of an analogue of a higher rank lattices in a Gromov hyperbolic gr…
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 group has an amenable action on the metric boundary of its Cayley graph. This result has significance for the study of Lip-norms on group C*-algebras.
Given a Gromov-hyperbolic group $G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of $G$. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry--Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms alo…
The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of this result. The first one is by a characterization of rank-$1$ isometries by Hamenstadt. The second proof follows directly from some results of Dahmani-Guirardel-Osin and Sisto. Then using Mihailova's construction, we show the existence of a finitely generated subgroup of the direct product of two free groups $\mathbb F_2\times \mathbb F_2$ without the bound…
Consider an ergodic non-singular action $Γ\cc B$ of a countable group on a probability space. The type of this action codes the asymptotic range of the Radon-Nikodym derivative, also called the {\em ratio set}. If $Γ\cc X$ is a pmp (probability-measure-preserving) action, then the ratio set of the product action $Γ\cc B\times X$ is contained in the ratio set of $Γ\cc B$. So we define the {\em stable ratio set} of $Γ\cc B$ to be the intersection over all pmp actions $Γ\cc X$ of the ratio sets of $Γ\cc B\times X$. By analogy, there is a notion of {\em stable type} which codes the stable ratio se…
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) <= 2. When d(S) >= 3 the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of R^k, k >= 2); when d(S) <= 2 we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of CP(S) prove Gromov-hyperbolicity. W…
Tits Alternative in groups with proper product actions on proper Gromov-hyperbolic spaces
本文研究具有(PPH)性质的群，证明其有限生成子群要么可算要么含$F_2$，并探讨具有(PPT)性质的群，建立超几何群在正则树乘积上的恰当作用条件，将Button的问题转化为非元素性超几何群是否具有(PPT)性质的判定。
We prove that for every finitely generated hyperbolic group $G$, the action of $G$ on its Gromov boundary induces a hyperfinite equivalence relation.
We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the…
We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct e.g. new groups having some interesting asphericity properties.
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 bicombings on this space. The first of these, "preferred paths", is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2πTheorem in the …
Random Quotients of Hyperbolic Spaces with Respect to WPD Actions and Linear Connectivity of the Boundary
该研究探讨两类双曲群的随机商，证明 acylindrically 双曲群在满足特定作用条件时，其随机元素正规闭包的商空间边界线性连通，分层双曲群对应商空间边界同胚于 Nöbeling 曲线。
We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm modules all represent the same, distinguished class in K-homology, namely that of the `boundary extension' of $C(\bd \G)\rtimes \G$ associated to the Gromov compactification of $\G$, and is typically nonzero. Their summability is closely related to the Hausdorff dimension of the boundary. We use these results to compute the Connes-Chern character of the bounda…
We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without $2$-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.
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