Square complexes and simplicial nonpositive curvature

Abstract:

We prove that each nonpositively curved square $\mathcal {VH}$-complex can be turned functorially into a locally $6$-large simplicial complex of the same homotopy type. It follows that any group acting properly and cocompactly on a CAT(0) square $\mathcal {VH}$-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 non-$\mathcal {VH}$ nonpositively curved square complex whose fundamental group is neither systolic nor even virtually systolic.