In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$ and suppose that there exists an element $g_0\in G$ such that (1) $Z_{g_0}$ is finite, (2) $X\setminus F_{g_0}$ is not connnected, and (3) each component of $X\setminus F_{g_0}$ is convex and not $g_0$-invariant, where $Z_{g_0}$ is the centralizer of $g_0$ and $F_{g_0}$ is the fixed-point set of $g_0$ in $X$ (that is, $Z_{g_0}=\{h\in G| g_0h=hg_0\}$ and $F_{g_0}=\{x\in X| g_0x=x\}$). Then we show that each orbit $G α$ is dense in the boundary $\partial X$ (i.e.\ $\partial X$ is minimal) and the set $\{g^{\infty} | g\in G, o(g)=\infty\}$ is also dense in the boundary $\partial X$. We obtain an application for dense subsets on the boundary of a Coxeter system.
We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\widetilde\sqcup(X_1,...,X_k)$, in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then show that, in various settings, the ideal boundary of the free product of groups is homeomorphic to the dense amalgam of boundaries of the factors. We give also related more general results for graphs of groups with finite edge groups. We justify these results by referring to a con…
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.
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.
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。
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.
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.
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 …
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…
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.
The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic-like behavior in the space. A key property of this boundary is that a quasi-isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper we investigate when the converse holds. We prove that for cocompact CAT(0) spaces, a homeomorphism of Morse boundaries is induced by a quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.
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.
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 …
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…
We show under weak hypotheses that the pushforward $\{Z_no\}$ of a random-walk to a CAT(0) cube complex converges to a point on the boundary. We introduce the notion of squeezing points, which allows us to consider the convergence in either the Roller boundary or the visual boundary, with the appropriate hypotheses. This study allows us to show that any nonelementary action necessarily contains regular elements, that is, elements that act as rank-1 hyperbolic isometries in each irreducible factor of the essential core.
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…
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…
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…
We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.
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…
We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of virtually fundamental groups of hyperbolic 3-manifolds. Keywords: CAT(0) cube complexes, Hyperbolic groups, hyperbolic 3-manifolds.
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 …
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…
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 .
We prove that an IRS of a group with a geometrically dense action on a CAT(0) space also acts geometrically densely; assuming the space is either of finite telescopic dimension or locally compact with finite dimensional Tits boundary. This can be thought of as a Borel density theorem for IRSs.
