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).

See also page on simplicial nonpositive curvature, maintained by T. Elsner including a list of 19 papers in systolic group theory

See also Swiatkowski's introduction to the subject from a 2007 conference

Prizes

Systolic arithmetic

Systolic topology

News and recent developments

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