Panos Papasoglu, Eric Swenson
We show that if a 1-ended group $G$ acts geometrically on a CAT(0) space $X$ and $\bd X$ is separated by $m$ points then either $G$ is virtually a surface group or $G$ splits over a 2-ended group. In the course of the proof we study nesting actions on $\R $-trees and we show that nesting actions with non overlapping translation intervals give rise to isometric actions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give concrete, "infinitesimal" conditions for a proper geodesically complete CAT(0) space to have semistable fundamental group at infinity.
Given a sublinear function $κ$, $κ$-Morse boundaries $\pka X$ of proper \CAT spaces are introduced by Qing, Rafi and Tiozzo. It is a topological space that consists of a large set of quasi-geodesic rays and it is quasi-isometrically invariant and metrizable. In this paper, we study the sublinearly Morse boundaries with the assumption that there is a proper cocompact action of a group $G$ on the \CAT space in question. We show that $G$ acts minimally on $\pka G$ and that contracting elements of $G$ induces a weak north-south dynamic on $\pka G$. Furthermore, we show that a homeomorphism $f \fro…
It is well known that the Tits boundary of a proper cocompact CAT(0) space embeds into every asymptotic cone of the space. We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e. cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. The resulting projection from any asymptotic cone to the Tits boundary is determined by the visual topology; on the …
On non-locally connected boundaries of CAT(0) spaces
本文研究了具有非局部连通边界的CAT（0）空间。给出了CAT（0）空间边界非局部连通的条件.
Tits rigidity of CAT(0) group boundaries
定义了CAT（0）群的视觉边界的Tits刚性，证明了两个Cantor集的并及其悬置是Tits刚性的.
In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of the boundaries are arbitrarily connected. We also prove a combination theorem for certain amalgams of CAT(0) groups to act on spaces with non-path connected visual boundaries. We apply this theorem to some right-angled Artin groups.
On equivariant homeomorphisms of boundaries of CAT(0) groups
本文研究了CAT（0）群$G$几何作用于其上的两个真CAT（0）空间$X$和$Y$的边界$\partial X$和$\partial Y$的等变同胚.我们给出了一个充分条件，使两个边界$\partial X$和$\partial Y$的$G$ -等变同胚成为由$ϕ(gx_0)=gy_0$定义的拟等距$ϕ:Gx_0\to Gy_0$的连续扩张，其中$x_0\in X$和$y_0\in Y$ .
Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one universal cover contains a nonplanar graph, while the visual boundary of the other does not. In contrast, we prove for any two locally CAT(0) metrics on the Croke-Kleiner complex, if a finite graph embeds in the visual boundary of one universal cover, then the graph embeds in the visual boundary of the other.
CAT(0) groups and Coxeter groups whose boundaries are scrambled sets
本文研究了边界为乱集的CAT（0）群和Coxeter群。假设一个群$G$在CAT（0）空间$X$上作几何作用（即通过等距适当地和余紧地）。(Such组$G$称为{ \it CAT（0）组}。）然后群$G$通过同胚作用在$X$的边界$\partial X$上，并且我们可以在边界$\partial X$上定义一个度量$d_{\partial X}$。如果对于具有$α\neqβ$的任何$α,β\in\partial X$，（1）$\limsup\{d_{\partial X}(gα,gβ) | g\in G\}>0$和（2）$\liminf\{d_{\partial X}(gα,gβ) | g\in G\}=0$，则边界$\partial X$被称为{ \it加扰集}。我们研究了CAT（0）群（和Coxeter群）的边界是什么时候.
All CAT(0) Boundaries of a Group of the Form HxK are CE Equivalent
M. Bestlett证明了对于任意给定的无挠CAT（0）群G，它的所有边界都是形状等价的。然后，他提出了一个问题，即它们是否满足更强的条件，即它们是细胞状等价物。在这篇文章中，我们证明了答案是“是的”的情况下，在该小组的问题分裂为一个直积与无限的因素。我们通过证明形状理论中一个有趣的定理来实现这一点。
Random Walks and Boundaries of CAT(0) Cubical complexes
我们证明了在弱假设下，随机游动到CAT（0）立方复形的前推$\{Z_no\}$收敛到边界上的一点。我们引入了挤压点的概念，这使我们能够考虑在辊边界或视觉边界的收敛，与适当的假设。这项研究使我们能够表明，任何非初等行动必然包含正规元素，也就是说，元素，作为秩-1双曲等距在每个不可约因子的本质核心。
Semistability and CAT(0) Geometry
我们解释了为什么单端真CAT（0）空间的半稳定性可以由测地线确定。这适用于CAT（0）组的边界。
On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter groups
本文研究了CAT（0）群$G$几何作用于其上的两个真CAT（0）空间$X$和$Y$的边界$\partial X$和$\partial Y$的等变同胚.给出了边界$\partial X$和$\partial Y$的$G$ -等变同胚是由$ϕ(gx_0)=gy_0$定义的拟等距$ϕ:Gx_0\rightarrow Gy_0$的连续扩张的充分条件和等价条件，其中$x_0\in X$和$y_0\in Y$ .本文称CAT（0）群$G$是{ \it等变（边界）刚性}的，如果$G$由上述等变同胚确定其理想边界.作为应用，我们引入了（非）等变刚性CAT（0）群的一些例子，并证明了如果Coxeter群$W_1$和$W_2$作为反射群是等变刚性的，那么$W_1 * W_2$也是等变刚性的.我们还对某些CAT（0）群的边界的非刚性性提出了一个猜想。
We prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $Γ$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of $\partial X$ equals $π$ and $Γ$ does not act minimally on $\partial X$, then $\partial X$ is a spherical building or a spherical join. If $X$ is also geodesically complete, then $X$ is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of $\partial X$, doe…
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.
Dense subsets of boundaries of CAT(0) groups
本文研究了CAT（0）群的边界的稠密子集。设群$G$几何地作用在CAT（0）空间$X$上，并设存在元素$g_0\in G$使得（1）$Z_{g_0}$是有限的，（2）不连接$X\setminus F_{g_0}$，以及（3）$X\setminus F_{g_0}$的每个分量是凸的并且不是$g_0$不变的，其中，$Z_{g_0}$是$g_0$的中心化器，$F_{g_0}$是固定的$X$中$g_0$点集（即$Z_{g_0}=\{h\in G| g_0h=hg_0\}$和$F_{g_0}=\{x\in X| g_0x=x\}$）。然后我们证明了每个轨道$G α$在边界$\partial X$（即\ $\partial X$是最小的）并且集合$\{g^{\infty} | g\in G, o(g)=\infty\}$在边界$\partial X$中也是稠密的。我们得到了一个应用程序的边界上的Coxeter系统的稠密子集。
Sphere boundaries of hyperbolic groups
本文证明了无穷远单连通双曲群$G$具有足够的余维1曲面子群，且有$\partial G \cong \mathbb{S}^2$ .结合Markovic的一个结果，我们的结果给出了双曲三维流形的虚基本群的一个新的刻画。关键词：CAT（0）立方复形，双曲群，双曲三维流形。
Coarse decompositions of boundaries for CAT(0) groups
在这项工作中，我们引入了一个新的组合概念的边界$\Re C$的$ω$维立方$C$。定义$\Re C$为$C$的标准半空间系上的几乎相等超滤子类的集合，并赋予类的Tychonoff闭包之间的相互作用的序关系.当$C$出现的对偶的一个cutilum-或离散系统的半空间-$\HH$的CAT（0）空间$X$（例如，Niblo-Reeves cutilum的戴维斯-Moussong复杂的有限秩Coxeter群），我们展示了如何$\HH$诱导一个功能$ρ:\bd X\to\Re C$。我们开发了一个概念的一致性$\HH$，推广了Coxeter群所享有的平行壁属性，并表明，如果对$(X,\HH)$承认一个几何行动的一组$G$，然后纤维的$ρ$形成分层的$\bd X$分级的顺序结构的$\Re C$。我们还展示了这种结构如何计算$X$的山雀边界的组件。最后，利用我们在另一篇文章中的结果，即一个群的一致性蕴涵着$C$的局部有限性，我们给出了$G$作用在$C$上的$ρ$的余紧性的一个条件，推广了Williams的一个结果，以前只知道对Coxeter群。
Examples of Non-Rigid CAT(0) Groups from the Category of Knot Groups
C Croke和B Kleiner构造了一个具有多个可视边界的CAT（0）群的例子。威尔逊证明了这个群有无数个不同的边界。本文证明了两个非平凡环面纽结的任何连通和的纽结群也有无数个不同的CAT（0）边界。
We show that if G is an admissible group acting geometrically on a CAT(0) space X, then G is a hierarchically hyperbolic space and with mild assumptions the sublinearly-Morse boundary of the group is a topological model for associated Poisson boundaries .
Boundaries and JSJ decompositions of CAT(0)-groups
设G是CAT（0）空间X上的单端离散余紧群。我们表明，X的边界没有割点，可以检测分裂的$G$在两端组和恢复其JSJ分解从边界。我们证明了CAT（0）空间X上群G的任何离散作用都满足收敛型性质。这是用于证明上述结果，但它也是独立的利益。特别地，如果G在X上作用余紧，那么作为推论，如果X的边界的Tits直径大于$\frac {3π} 2$，那么它是无限的，并且G包含秩为2的自由子群。
The Action of Thompson's Group on a CAT(0) Boundary
证明汤普森群F不顺从的一种方法是在局部紧CAT（0）空间X上展示F的作用，该空间X不包含F-不变平坦且在无穷远处的边界上不存在全局不动点。本文研究了Thompson群F、T和V在真CAT（0）立方复形无穷远边界上的作用。特别地，我们证明了Thompson群T和V的行为在它们的CAT（0）立方复形的边界上没有固定任何点。这尤其再次证明了一个众所周知的事实，即这些群体是不服从的。我们得到了F的不动集的一个部分描述：Thompson群F在其三次复形的边界上固定了一条弧。我们保留有更多不动点的可能性，但描述一个必须包含所有其他不动点的边界区域。
The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X is locally finite, essential, irreducible, non-Euclidean and admits a cocompact action by a group G, Nevo-Sageev show that a subset, B(X), of the Roller boundary is the realization of the Poisson boundary and that the action of G on B(X) is minimal and strongly proximal. Additionally, these authors show B(X) satisfies many other desirable dynamical and topological properties. In this article we give several equivalent characterizations for when B(X) is equal to the entire Roller boundary. As an application …
Boundaries of systolic groups
对于所有的收缩群，我们构造的边界是EZ-结构。这暗示了对无无挠的收缩群的Novikov猜想。边界是通过一个系统的区别测地线在收缩复合体，我们证明了有粗略的相似性质的测地线在CAT（0）空间。
We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of ${\rm CAT(0)}$ cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group.
A rank-one CAT(0) group is determined by its Morse boundary
适当的测地线度量空间的Morse边界被设计成在空间中编码双曲状行为。这个边界的一个关键性质是两个这样的空间之间的准等距在它们的莫尔斯边界上引起同胚。在本文中，我们调查时，逆持有。证明了在余紧CAT（0）空间中，Morse边界的同胚是由拟等距诱导的当且仅当该同胚是拟mobius和2-稳定的.
This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition.
Generalizing the Croke-Kleiner Construction
众所周知，每个词的双曲群都有一个明确的视觉边界。以C. Croke和B. Kleiner表明，对于CAT（0）群，情况并非如此。然而，正如M所观察到的，CAT（0）群的所有边界都是形状等价的。Besteland和R.纪勤Bestlaughter问它们是否也满足更强的条件，即它们是细胞样等价物。本文描述了一种构造方法，它将产生具有多个边界的CAT（0）组。这些群在高维中有非常复杂的边界。我们希望，他们的研究可以为Bestlew的问题提供见解。
Let $X$ be a nonelementary CAT(0) cubical complex. We prove that if $X$ is essential and irreducible, then the contact graph of $X$ (introduced in \cite{Hagen}) is unbounded and its boundary is homeomorphic to the regular boundary of $X$ (defined in \cite{Fernos}, \cite{KarSageev}). Using this, we reformulate the Caprace-Sageev's Rank-Rigidity Theorem in terms of the action on the contact graph. Let $G$ be a group with a nonelementary action on $X$, and $(Z_n)$ a random walk corresponding to a generating probability measure on $G$ with finite second moment. Using this identification of the bou…
Quasi-redirecting boundaries of non-positively curved groups
本文证明多类群的拟重定向边界存在，刻画Croke-Kleiner容许群的边界并证明其满射到Bowditch边界，拓宽了该边界的存在性。
On boundaries of Coxeter groups and topological fractal structures
本文基于W.Ballmann和M.Brin关于秩一等距的研究以及最近P.Caprace和K.Fujiwara关于Coxeter群的秩一等距的研究，研究了Coxeter群边界的拓扑分形结构.我们还证明了Coxeter群的边界上的极限点集是稠密的，并对秩为1的等距CAT（0）群的边界作了一些讨论。
Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
设G是几何作用在CAT（0）立方复形X上的群.我们首先证明了G是相对于子群集合P的双曲的当且仅当X的单纯边界是一个非空的离散集的不交并，以及一个在适当意义下对应于P的元素的子复形的成对不交集合。Hruska-Kleiner关于相对双曲CAT（0）空间的Tits边界的一个定理.其次，我们将立方复形X的渐近锥面中截点的存在性与X的边界的1-骨架的有界性联系起来。利用群G在CAT（0）立方复形X上的边界作用的结构和性质，给出了群G在CAT（0）立方复形X上正常作用和余紧作用的厚和强代数厚的特征.最后，对每个n，k，我们构造了无穷多个群G的拟等距类型，使得G是n阶强代数厚的，具有n+1阶多项式发散，并且在k维CAT（0）立方复形上正确地和余紧地作用.
Relations between various boundaries of relatively hyperbolic groups
假设一个群$G$相对于它的子群的集合$\PP$是双曲的，并且在一个$\CAT(0)$（或$δ$ -双曲）空间$X$上也适当地起作用，协紧。相对双曲结构提供相对边界$\partial(G,\PP)$。$\CAT(0)$结构在无穷远处提供了不同的边界$\partial X$。在这篇文章中，我们将研究这两个空间在无穷远处的联系。特别是，我们表明，$\partial (G,\PP)$是$G$ -等变同胚的空间从$\partial X$通过确定周边极限点相同类型的。
Homotopy equivalent boundaries of cube complexes
本文研究有限维 CAT(0) 立方体复形的多种边界，通过 Roller 边界商上的偏序构造单纯 Roller 边界，并证明其与 Tits、单纯边界在同伦意义下 $Aut(X)$-等变等价，从而说明度量扰动不改变等变同伦类型。
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…
Hyperbolic groups with planar boundaries
证明了凸余紧Kleinian群类是拟等距刚性的。我们还建立了一个词双曲群的平面边界不同于球实际上是一个凸余紧克莱因群，如果它的边界有Ahlfors正则共形维数严格小于$2$或如果它的几何作用在CAT（0）立方复形。
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…
Local Connectivity of Right-angled Coxeter group boundaries
本文研究直角 Coxeter 群所作用 CAT(0) 空间边界的局部连通性，给出由定义图刻画的充分条件，并对已发表版本进行证明精简与插图补充。
$E\mathcal{Z}$-boundaries, splittings over finite subgroups, and dense amalgams
本文研究了$E\mathcal{Z}$-边界框架中无限端群的边界在有限子群分裂下的形式，证明其可表示为因子子群的极限集的密集 amalgam。
In this paper, we study boundary actions of CAT(0) spaces from a point of view of topological dynamics and $C^*$-algebras. First, we investigate the actions of right-angled Coexter groups and right-angled Artin groups with finite defining graphs on the visual boundaries and the Nevo-Sageev boundaries of their natural assigned CAT(0) cube complexes. In particular, we establish (strongly) pure infiniteness results for reduced crossed product $C^*$-algebras of these actions through investigating the corresponding $\cat$ cube complexes and establishing necessary dynamical properties such as minima…
To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $κ$, we consider a subset of the visual boundary called the $κ$-Morse boundary and show that …
CAT(0) and CAT(-1) fillings of hyperbolic manifolds
我们给新的例子双曲和相对双曲群的上同调维数$d$的所有$d\geq 4$。这些例子是将CAT $(0)$ /CAT $(-1)$填充构造（基于奇异双翘曲积）应用于具有toral尖点的有限体积双曲流形的结果。得到的群有一些有趣的性质，这些性质是通过分析它们在无穷远处的边界而建立的，这些边界是由一种莫尔斯理论技术建立的，这种技术与普通的和组合的莫尔斯理论有关但又不同。
On boundaries of bicombable spaces
本文系统研究了作用于一致锥形测地双梳空间上的群的EZ-结构，构造了此类结构并证明其同胚唯一性，同时给出了与CAT(0)空间类似的边界性质。
Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras
本文通过验证 Myrberg 点的自由性，证明 CAT(0) 空间的多类边界作用（视觉、horofunction、Roller）是拓扑自由的强边界作用，推广了已知结果，并应用于构造新的 $C^*$-无自群及精确纯无限单的约化交叉积 $C^*$-代数。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。