A combinatorial non-positive curvature I: weak systolicity

We introduce the notion of weakly systolic complexes and groups, and initiate regular studies of them. Those are simplicial complexes with nonpositive-curvature-like properties and groups acting on them geometri-cally. We characterize weakly systolic complexes as simply connected sim-plicial complexes satisfying some local combinatorial conditions. We provide several classes of examples — in particular systolic groups and CAT(-1) cubical groups are weakly systolic. We present applications of the theory, concerning Gromov hyperbolic groups, Coxeter groups and systolic groups