CAT(0) square complexes that do not embed into finite products of trees

RESUMO

Answering a question of Chepoi and Hagen, we give two constructions of bounded degree CAT(0) square complexes that cannot be isometrically embedded into any finite product of trees. The first is based on Burling graphs and has degree at most five, which is optimal. These square complexes topologically embed into $\R^3$ since we further prove that every CAT(0) square complex whose vertices have degree at most five can be topologically embedded into the product of a line and a star. It turns out that in this setting, Burling graphs are the only obstruction: we prove that CAT(0) square complexes that topologically embed into $\R^3$ and whose crossing graph forbids some induced Burling graph can be isometrically embedded into a finite product of trees. Our second CAT(0) square complex has degree at most six and its crossing graph forbids an induced Burling graph. Of independent graph-theoretic interest, along the way we construct intersections of chordal and interval graphs with arbitrarily large girth and chromatic number. We also discuss connections to nice labellings of event structures.