Shubhabrata Das, Mahan Mj
We prove the existence of Cannon-Thurston maps for Kleinian groups corresponding to pared manifolds whose boundary is incompressible away from cusps. We also describe the structure of these maps in terms of ending laminations.
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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).
Hyperbolic boundaries vs. hyperbolic groups
该讲义将双曲群、相对双曲群理论与流形、克莱因群理论关联，给出相对双曲群及其边界的定义与实例，综述领域研究，内容源于2018年夏季CIRM的讲座。
Parabolic cut pairs in boundaries of relatively hyperbolic groups
本文针对相对双曲群边界中未被探索的抛物切割对现象，提出组合定理构造相关群示例并证明其完备性，同时给出两种适用的边界拓扑描述。
We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence spectra of relatively hyperbolic groups. We show the existence of an infinite collection of right-angled Coxeter groups which all have exponential divergence but they all have different divergence spectra. We also study Morse boundaries of relatively hyperbolic groups and examine their connection with Bowditch boundaries.
Quasiconformal Maps between Bowditch Boundaries of Relatively Hyperbolic Groups
研究相对双曲群的拟等距分类，核心方法是通过鲍迪奇边界引入拟共形映射概念，主要贡献是证明相对双曲群间粗保尖点的拟等距与粗保horoball阴影的鲍迪奇边界拟共形同胚相互诱导。
In this paper we prove that if some relatively hyperbolic groups have Bowditch boundary homeomorphic to the $n$-sphere, then they are also relatively hyperbolic with respect to another set of parabolic subgroups and its Bowditch boundary is homeomorphic to the $n-1$-dimensional Sierpiński carpet.
We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a 2-stable, quasi-möbius homeomorphism between their Morse boundaries. Our result extends a recent result of Charney-Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental group…
In this expository note, we illustrate phenomena and conjectures about boundaries of hyperbolic groups by considering the special cases of certain amalgams of hyperbolic groups. While doing so, we describe fundamental results on hyperbolic groups and their boundaries by Bowditch and Haissinsky, along with special treatments for the boundaries of free groups by Otal and Cashen.
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.
In this paper, we study the Hausdorff dimension of the Floyd and Bowditch boundaries of a relatively hyperbolic group, and show that for the Floyd metric and shortcut metrics respectively, they are are both equal to a constant times the growth rate of the group. In the proof, we study a special class of conical points called uniformly conical points and establish that, in both boundaries, there exists a sequence of Alhfors regular sets with dimension tending to the Hausdorff dimension and these sets consist of uniformly conical points.
We show that the sublinearly Morse boundary of a CAT(0) cubical group with a factor system is well-defined up to homeomorphism with respect to the visual topology. The key tool used in the proof is a new topology on sublinearly Morse boundaries that is induced by group actions on hyperbolic spaces that are sufficiently nice, for example, largest acylindrical actions. Using the same techniques, we obtain a explicit description of this new topology on the sublinearly Morse boundary of any hierarchically hyperbolic group in terms of medians. Finally, we explicitly describe the sublinear Morse bou…
Visual metrics on boundaries of hyperbolic spaces
本文是基于2024年里弗赛德几何群论研讨会迷你课程的综述，介绍双曲度量空间边界可视化度量的构造、拟对称映射及不变量、相关维数、Gromov圆形树的构造与应用，并辅以示例。
In this paper we prove a theorem describing the local topology of the boundary of a hyperbolic group in terms of its global topology: the boundary is locally simply connected if and only if the complement of any point in the boundary is simply connected. This generalises a theorem of Bestvina and Mess.
We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
Topological stability of relatively hyperbolic groups acting on their boundaries
本文证明了双曲群在其Bowditch边界上作用的拓扑稳定性，指出在抛物子群上受半共轭控制的小扰动必全局半共轭于标准作用，从而确立了arXiv:2206.14914主要结果的相对版本。
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.
Maps between relatively hyperbolic spaces and between their boundaries
本文建立相对双曲群/空间间尊重外围结构的拟等距嵌入与边界拟对称嵌入的对应，并推广至多多项式失真情形，据此刻画相对虚拟幂零子群双曲的群。
We provide, for hyperbolic and flat 3-manifolds, obstructions to bounding hyperbolic 4-manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.
Local connectedness of boundaries for relatively hyperbolic groups
本文证明了相对单端的相对双曲群对的 Bowditch 边界具有局部连通性，去除了 Bowditch 先前结果中对外围子群的有限展示、端数及挠子群等全部额外限制。
Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $δ$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary $\partial(G,\PP)$. The $\CAT(0)$ structure provides a different boundary at infinity $\partial X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that $\partial (G,\PP)$ is $G$--equivariantly homeomorphic to the space obtained from $\partial X$ by identifying the peripheral limit points of the sam…
Stability of hyperbolic groups acting on their boundaries
本文利用边界点的动力学编码，证明双曲群在其 Gromov 边界上的作用具有拓扑稳定性，并为圆边界情形的全局稳定性提供新证明。
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 spaces as boundaries of arbitrary Coxeter groups with manifold nerves, and as Gromov boundaries of the fundamental groups of singular spaces obtained from some finite volume hyperbolic manifolds by cutting off their cusps and collapsing the resulting boundary tori to points.
Relatively hyperbolic groups with planar boundaries
本文在相对双曲群边界为平面且无 Sierpinski carpet 或割点、群单端且实质上无 2-挠的条件下，证明了 Martin-Skora 猜想的一个版本，即该群实质 Kleinian，并应用于 Cannon 猜想及 S^2 上的收敛群。
Maps between Boundaries of Relatively Hyperbolic Groups
本文研究了相对超几何群边界之间的映射，通过引入相对准 Mobius 映射的概念，证明了同胚与准等距之间的等价性。
Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the appropriate sense, to elements of P. As a special case of this result is a new proof, in the cubical case, of a Theorem of Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0) spaces. Second, we relate the existence of cut-points in asymptotic cones of a cub…
Poisson boundaries of building lattices and rigidity with hyperbolic-like targets
本文证明正则厚欧氏建筑及其格的泊松边界为无穷远房室空间并带调和测度类，进而推广了从建筑格到负曲率群的态射与上闭链的刚性结果。
In this paper, we develop a theory on the degenerations of Blaschke products $\mathcal{B}_d$ to study the boundaries of hyperbolic components. We give a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component $\mathcal{H}_d$ containing $z^d$. We show the closure $\overline{\mathcal{H}_d}$ is not a topological manifold with boundary for $d\geq 4$ by constructing self-bumps on its boundary.
Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly connected" if and only if the boundary $\partial (G,\mathcal P)$ has no cut point. Our goal is to produce linearly connected metrics on $\partial (G,\mathcal P)$ that are "piecewise" visual when $\partial (G,\mathcal P)$ contains cut points. %Visual metrics for $\partial (G,\mathcal P)$ are tightly linked to inner products of geodesic rays in "cusped" spaces…
Given a complex of groups over a finite simplicial complex in the sense of Haefliger, we give conditions under which it is possible to build an EZ-structure in the sense of Farrell-Lafont for its fundamental group out of such structures for its local groups. As an application, we prove a combination theorem that yields a procedure for getting hyperbolic groups as fundamental groups of simple complexes of hyperbolic groups. The construction provides a description of the Gromov boundary of such groups.
Let $Γ$ be a relatively hyperbolic group and let $μ$ be an admissible symmetric finitely supported probability measure on $Γ$. We extend Floyd-Ancona type inequalities up to the spectral radius of $μ$. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on $Γ$ is stable in the sense of Picardello and Woess. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We prove that this criterion is always satisfie…
A Characterization of Relative Hyperbolicity via Morse and Contracting Boundaries
本文通过莫尔斯边界和收缩边界证明了相对双曲群的刻画，研究了非元素性的双曲性条件。
Fixed-point-free involutions on the boundaries of Right-angled Coxeter Groups
本文证明 Nica 关于边界存在无不动点对合的猜想在直角 Coxeter 群情形下的变体，并为原词双曲群问题提供方法启发。
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathcal X$ is proper, we get a compactification of $\mathcal X$; we also prove that our construction generalizes the Gromov boundary of a hyperbolic space. In our first main set of applications, we introduce a notion of geometrical finiteness for hierarchically hyperbolic subgroups of…
We study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. We show that if $(G,\mathcal{P})$ is a rigid relatively hyperbolic group pair whose boundary embeds in $S^2$, then the action on the boundary extends to a convergence group action on $S^2$. More generally, if the boundary is connected and planar with no cut points, we show that every element of $\mathcal{P}$ is virtually a surface group. This conclusion is consistent with the conjecture that such a group $G$ is virtually Kleinian. We give numerous examples to show the nece…
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner p…
We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the Poincare metric when the domain is uniformly perfect. We also establish Marden and Markovic's conjecture that a hyperbolic domain is uniformly perfect if and only if the nearest point retraction is Lipschitz with respect to the Poincare metric.
In the paper, we prove that a Moran set is homeomorphic to the hyperbolic boundary of the representing symbolic space in the sense of Gromov, which generalizes the results of Lau and Wang [Indiana U. Math. J. {\bf 58} (2009), 1777-1795]. Moreover, by making use of this, we establish the Lipschitz equivalence of a class of Moran sets.
On the Čech cohomology of Morse boundaries
研究超曲面n维流形的Morse边界上的Čech上同调，证明实系数减去上同调在维度至多n-3时消失，在n-2维不消失，且适用于相对超曲面群。
A seminal result in geometric group theory is that a 1-ended hyperbolic group has a locally connected visual boundary. As a consequence, a 1-ended hyperbolic group also has a path connected visual boundary. In this paper, we study when this phenomenon occurs for CAT(0) groups. We show if a 1-ended CAT(0) group with isolated flats acts geometrically on a CAT(0) space, then the visual boundary of the space is path connected. As a corollary, we prove all CAT(0) groups with isolated flats are semistable at infinity.
The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic $n$-manifolds. An analogous result holds for non-compact orientable hyperbolic arithmetic $n$-manifolds of finite volume that are geometric boundaries, for $n \geq 2$.
On Uniformly Perfect Morse Boundaries
本文研究了proper geodesic metric spaces中均匀完美Morse边界的概念，并证明了非平凡群的Morse边界在非空时的均匀完美性，同时建立了相关边界的同胚刚性定理。
Ratio-limit boundaries for random walks on relatively hyperbolic groups
本文研究相对双曲群上随机游走的比率极限边界，扩展了Woess策略并揭示其局限，证明谱非退化情形下极小点闭包为唯一最小不变子空间，并推广了Toeplitz C*-代数共泛商的存在性，且构造了反例。
In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.
On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension
本文将Bonk-Kleiner定理推广到相对双曲群情形，证明若相对双曲群的Bowditch边界是满足条件的Ahlfors正则2-球面，则该群在双曲空间ℍ³上离散等距作用，且其相对子群为几乎ℤ²。
Relative hyperbolicity, thickness, and the hierarchically hyperbolic boundary
本文利用分层双曲边界的单纯结构刻画了HHG的相对双曲性与1阶厚度，并证明相对双曲HHG的Bowditch边界可由HHS边界坍缩外围子群极限集得到，同时给出纳入双曲嵌入子群以修改HHG结构的构造。
(Non-)existence of Cannon-Thurston maps for Morse boundaries
本文研究正规子群在Morse边界上Cannon-Thurston映射的存在性问题，证明其与双曲情形不同，可同时出现存在与不存在的现象。
Extensions of multicurve stabilizers are hierarchically hyperbolic
本文证明亏格至少为2的闭可定向曲面上多曲线稳定子的曲面群扩张为分层双曲群，回答了相关公开问题，并在附录中刻画一类分层双曲群的Morse边界。
Critical Exponents on Hyperbolic Surfaces with Long Boundaries and the Asymptotic Weil-Petersson Form
该研究以归一化Weil-Petersson测度为概率测度，利用脊图构造对比双曲曲面与度量缎带图模空间的临界指数随机变量，证明长边界下的渐近均值收敛及测度拉回的定量一致估计。
We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.
We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.
Discrete embeddings of hyperbolic groups with Pontryagin-surface boundaries
该研究构造了神经与$X_p$同胚的双曲直角Coxeter群，将其离散忠实凸紧嵌入到Isom($\mathbf H^5$)，证明五维最优且$p=2,3$时得循环对称无限族。
Hyperbolic Handlebody Complements in 3-Manifolds
本文证明紧致可定向3-流形经边界处理后包含任意亏格柄体且其补空间闭包为双曲流形，并通过扩展八面体分解得到部分柄体补的体积界限。
Given a compact orientable 3-manifold M whose boundary is a hyperbolic surface and a simple closed curve C in its boundary, every knot in M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which the geodesic representative of C is as small as you like.
We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d\geq 4$. These examples result from applying CAT$(0)$/CAT$(-1)$ filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory.
A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries
本文通过树映射证明了具有边界的亏格为0的双曲表面的Weil-Petersson体积为边界长度的多项式函数，并建立了距离依赖的三点函数公式。
Tukia and Vaisala showed that every quasi-conformal map of $\R^n$ extends to a quasi-conformal self-map of $\R^{n+1}$. The restriction of the extended map to the upper half-space $\R^n \times \R^+$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every homogeneous negatively curved manifold decomposes as $M = N \rtimes \R^+$ where $N$ is a nilpotent group with a metric on which $\R^+$ acts by dilations. We show that under some assumptions on $N$, every quasi-symmetry of $N$ extends to a bi-Lipschitz map of $M$. The result applies to a wide class of manifolds $M$…
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…
Conformal dimension bounds for certain Coxeter group Bowditch boundaries
针对定义图为完全图且边标签≥3的Coxeter群，通过拟等距嵌入圆树和CAT(-1)空间的几何有限作用给出其Bowditch边界共形维数的上下界，揭示相关群族存在无限拟等距类并得到共形维数的稠密性结果。
For a proper geodesic metric space $X$, the Morse boundary $\partial_*X$ focuses on the hyperbolic-like directions in the space $X$. It is a quasi-isometry invariant. That is, a quasi-isometry between two hyperbolic spaces induces a homeomorphism on their boundaries. In this paper, we investigate additional structures on the Morse boundary $\partial_*X$ which determine $X$ up to a quasi-isometry. We prove that, for $X$ and $Y$ proper, cocompact spaces, a homeomorphism $f$ between their Morse boundaries is induced by a quasi-isometry if and only if $f$ and $f^{-1}$ are bihölder, or quasi-symmet…
Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space
研究双曲三维空间中非紧凸集的刚性，证明若两个非紧闭凸子集的理想边界为圆型闭集且连通分支可数，则边界间的内蕴等距可延拓为整体等距，推广了Luo-Luo-Rao的刚性定理，并建立了Weyl问题的唯一性结果。
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