Let G be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex X without a global fixed point at infinity. We show that for any finite collection of simultaneously inessential subgroups {H1, . . . , Hk} in G, there exists an element g of infinite order such that ∀i, 〈Hi, g〉 ∼= Hi ∗ 〈g〉. We apply this to show that any group, acting faithf...

Let X, d be a metric space and x, y ∈ X with l d x, y . A geodesic path from x to y is an isometry c : 0, l → X such that c 0 x and c l y. The image of a geodesic path is called a geodesic segment. A metric spaceX is a (uniquely) geodesic space if every two points ofX are joined by only one geodesic segment. A geodesic triangle x1, x2, x3 in a geodesic space X consists of three points x1, x2, x...

In this paper, we study the convergence of implicit Mann iteration processes with bounded perturbations for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the -convergence results of implicit Mann iteration schemes with bounded perturbations for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the s...

We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that G is a group for which there is a bound on the orders of its finite subgroups. We prove that if G acts properly on a finitedimensional CAT(0) cubical complex, then either G contains a free subgroup of rank 2 or G is finitely generated and virtually abelian. In particular the above conclusion ...

We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)metric, whose universal covers Q M satisfy Hruska’s isolated flats condition, and contain 2-dimensional flats F with the property that @1F Š S ,! S Š @1 Q M are nontrivial knots. As a consequence, we obtain that the group 1.M/ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonposit...