7‐Location, weak systolicity, and isoperimetry

Abstract ‐Location is a local combinatorial condition for flag simplicial complexes introduced by Osajda. Osajda showed that simply connected 8‐located locally 5‐large complexes are hyperbolic. We treat the nonpositive curvature case of 7‐located locally 5‐large complexes. We show that any minimal area disc diagram in a 7‐located locally 5‐large complex is itself 7‐located and locally 5‐large. We define a natural metric for 7‐located disc diagrams and use this to prove that simply connected 7‐located locally 5‐large complexes have quadratic isoperimetric function. Along the way, we prove that locally weakly systolic complexes are 7‐located locally 5‐large.