Related papers: Comparing the Roller and B(X) boundaries of CAT(0)…

We prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite. This can be considered as a generalization of…

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of ${\rm CAT}(0)$ cube complexes. Examples of median spaces with compact intervals include all finite rank median spaces and all proper…

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ \bar{X} = X \cup \partial X$. Using this fact, several results known for proper…

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…

M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we…

Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that the boundary of X has no cut points and that one can detect splittings of $G$ over two-ended groups and recover its JSJ decomposition from the…

We show that an automorphism of an arbitrary CAT(0) cube complex either has a fixed point or preserves some combinatorial axis. It follows that when a group contains a distorted cyclic subgroup, it admits no proper action on a discrete…

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…

Let $G$ be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex $X$ without fixed points at infinity. We show that for any finite collection of simultaneously inessential subgroups…

In this paper, we study topological dynamics on the visual boundary and several combinatorial boundaries associated to $\operatorname{CAT}(0)$ spaces. Through verifying the freeness of Myrberg points on the boundaries, we prove that a large…

Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the…

The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups. In this paper, we establish that the QR boundary exists as a topological space for several well-studied classes of…

Let G be a locally compact group, let X be a universal proper G-space, and let Z be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup H of G. Let W be the resulting boundary. Assuming the…

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group $G$ acting geometrically on a CAT(0) space $X$ we show there is a flat $F\subset X$ of maximal dimension whose boundary sphere intersects every minimal…

Let $X$ be a proper geodesic metric space and let $G$ be a group of isometries of $X$ which acts geometrically. Cordes constructed the Morse boundary of $X$ which generalizes the contracting boundary for CAT(0) spaces and the visual…

Let B be the complex n-dimensional ball and X' be the toroidal compactification of a quotient B/G by a torsion free lattice G of SU(n,1). For an arbitrary G-rational boundary point p, denote by U(p) the commutant of the unipotent radical of…

We prove that if $G = G_1\times\dots\times G_n$ acts essentially, properly and cocompactly on a CAT(0) cube complex X, then the cube complex splits as a product. We use this theorem to give various examples of groups for which the minimal…

We exhibit a variety of groups that act properly and even cocompactly on median graphs (a.k.a. one-skeletons of CAT(0) cube complexes), with quasi-isometric groups that do not admit any proper action on a median graph. This answes a…

We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on…

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…