Quasi-isometry invariants of weakly special square complexes

We define the intersection complex for universal cover of a compact weakly special square and show that it is quasi-isometry invariant. By using this invariant, we study quasi-isometric classification 2-dimensional right-angled Artin groups planar graph 2-braid groups. Our results two well-known cases groups: (1) those whose defining graphs are trees (2) outer automorphism finite. Finally, there infinitely many which to not.