Square complexes and simplicial nonpositive curvature

TL;DR

This paper demonstrates how nonpositively curved square VH-complexes can be transformed into simplicial complexes with nonpositive curvature properties, establishing systolicity of their fundamental groups and providing counterexamples.

Contribution

It introduces a functorial method to convert square VH-complexes into simplicial complexes, proving systolicity of groups acting on them and presenting a counterexample for non-VH complexes.

Findings

Finitely generated free groups' product is systolic.

Non-VH nonpositively curved square complexes can have non-systolic fundamental groups.

A functorial transformation preserves homotopy type and curvature properties.

Abstract

We prove that each nonpositively curved square VH-complex can be turned functorially into a locally 6-large simplicial complex of the same homotopy type. It follows that any group acting geometrically on a CAT(0) square VH-complex is systolic. In particular the product of two finitely generated free groups is systolic, which answers a question of Daniel Wise. On the other hand, we exhibit an example of a compact non-VH nonpositively curved square complex, whose fundamental group is neither systolic, nor even virtually systolic.