7-location, weak systolicity and isoperimetry

TL;DR

This paper investigates 7-located, locally 5-large complexes, establishing their nonpositive curvature properties, quadratic isoperimetric inequalities, and connections to weak systolic complexes, extending Osajda's work on hyperbolicity.

Contribution

It proves that 7-located, locally 5-large complexes have quadratic isoperimetric functions and introduces a CAT(0) metric for disc diagrams, advancing understanding of their geometric properties.

Findings

Minimal area disc diagrams are 7-located and locally 5-large.

7-located complexes have quadratic isoperimetric functions.

Locally weakly systolic complexes are 7-located and locally 5-large.

Abstract

$m$-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 CAT(0) 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.