Special Topics

  AI In mathematics, systolic groups are groups that act geometrically (properly discontinuously and cocompactly) on a systolic complex . These groups are a primary focus in geometric group theory because they exhibit a form of "simplicial non-positive curvature," making them combinatorial analogs of CAT(0) or non-positively curved spaces. [ 1 , 2 , 3 , 4 , 5 ] Key Components Systolic Complex : A connected, simply connected simplicial complex where the "links" of all its simplices are 6-large . 6-large means the complex is "flag" (any set of vertices that are pairwise connected by edges spans a simplex) and has no embedded cycles of length less than 6 that are "full subcomplexes". Essentially, every simplicial loop of length 4 or 5 must have a diagonal. Geometric Action : For a group to be called systolic, it must act by simplicial automorphisms on one of these complexes such that the action is: Proper : Each compact subcomplex is moved away f...