Systolic complexes are simplicial analogues of nonpositively curved cubical complexes. T. Januszkiewicz and J. Świątkowski studied their asymptotic filling properties and showed that many groups acting on them geometrically are very different from classical ones. In particular, they proved the existence in arbitrary cohomological dimension of hyperbolic groups that contain no subgroups isometric to fundamental groups of nonpositively curved Riemannian manifolds of dimension greater than 2.
Given an integer k>3, a simplicial complex is called locally k-large if every link has the following property: every cycle consisting of fewer than k edges in the link, necessarily contains a pair of consecutive edges contained in a common 2-simplex of the link.
A simplicial complex is k-systolic if it is locally k-large, connected, and simply connected. Then 6-systolic and 7-systolic are related respectively to CAT(0) and CAT(-1).
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