With some exceptions in Chapter IV the translation follows the Annals version of the article. We included and updated only these references, on which Chapters IVâ€“VI rely. We added some comments asâ€¦ (More)

We prove that on a punctured oriented surface with Euler characteristic Ï‡ < 0, the maximal cardinality of a set of essential simple arcs that are pairwise nonhomotopic and intersecting at most onceâ€¦ (More)

An alternative definition of a special group is the following. A nonpositively curved cube complex X is special if its immersed hyperplanes do not self-intersect, are two-sided, do not directlyâ€¦ (More)

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in Ï€1M , and that the double cosets for crossing surfaces are also separable. Weâ€¦ (More)

We describe unicorn paths in the arc graph and show that they form 1â€“slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7â€“hyperbolic. Considering the same pathsâ€¦ (More)

We prove that for an action of a finite group G on a systolic complex X there exists a Gâ€“invariant subcomplex of X of diameter 5. For 7â€“systolic locally finite complexes we prove there is a fixedâ€¦ (More)

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on simplicial trees. This implies that their Gromov boundaries,â€¦ (More)

We prove a theorem conjectured by D.T. Wise in [9], that if a group acts properly discontinuously and cocompactly on a systolic complex, in whose 1â€“skeleton there is no isometrically embedded copy ofâ€¦ (More)