Given a CAT(0) cube complex X, we show that if Aut(X) 6= Isom(X) then there exists a full subcomplex of X which decomposes as a product with R. As applications, we prove that ifX is δ-hyperbolic, cocompact and 1-ended, then Aut(X) = Isom(X) unless X is quasi-isometric to H, and extend the rank-rigidity result of Caprace–Sageev to any lattice Γ ≤ Isom(X).

For a Jordan domain in the plane the length metric space of points connected to an interior point by a curve of finite length is a cat(0) space. With respect to the cone topology, that space plus its boundary at infinity is topologically the same as the original Jordan domain.

In this article we study the curvature properties of the order complex of a graded poset under a metric that we call the “orthoscheme metric”. In addition to other results, we characterize which rank 4 posets have CAT(0) orthoscheme complexes and by applying this theorem to standard posets and complexes associated with four-generator Artin groups, we are able to show that the 5-string braid gro...

It is a long standing open problem wether or not any word hyperbolic group admits a discrete faithful cocompact isometric action on a space of negative curvature. The goal of this note is to show that the answer is negative if one restricts to the class of groups of isometries of 2-dimensional CAT(0)-complexes. Namely, we will prove the following: Theorem 0.1 There exists a word-hyperbolic grou...

We obtained sufficient conditions for existence of fixed points of involutions in CAT(0) spaces. Convergence results of Mann and Ishikawa iterates of weakly contractive mappings are also proved.

We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.

In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a...

It is shown that certain natural maps between the ideal, Gromov, and end boundaries of a complete CAT(0) space can fail to be either injective or surjective. Additionally the natural map from the Gromov boundary to the end boundary of a complete CAT(−1) space can fail to be either injective or surjective.

We investigate the Tits boundary of locally compact CAT (0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the 2-complex and for a group generated ...