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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 IV–VI rely. We added some comments as… (More)
We prove that on a punctured oriented surface with Euler characteristic χ < 0, the maximal cardinality of a set of essential simple arcs that are pairwise nonhomotopic and intersecting at most once… (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 directly… (More)
Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in π1M , and that the double cosets for crossing surfaces are also separable. We… (More)
We describe unicorn paths in the arc graph and show that they form 1–slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7–hyperbolic. Considering the same paths… (More)
We prove that for an action of a finite group G on a systolic complex X there exists a G–invariant subcomplex of X of diameter 5. For 7–systolic locally finite complexes we prove there is a fixed… (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,… (More)
We prove that if S is a properly embedded π1-injective surface in a compact 3-manifold M , then π1S is separable in π1M .
We prove that the ending lamination space of the five-punctured sphere is homeomorphic to the Nöbeling curve.
We prove a theorem conjectured by D.T. Wise in , that if a group acts properly discontinuously and cocompactly on a systolic complex, in whose 1–skeleton there is no isometrically embedded copy of… (More)